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 Logic

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2024-07-29 -08:31 -0700

  
see also:
Readings in Theory of Computation
Readings in Philosophy
Readings in Mathematics
    
[Barendregt1977]
Barendregt, Henk P.  The Type Free Lambda Calculus.  Chapter D.7 in The Handbook of Mathematical Logic, Jon Barwise, editor.  Elsevier (Amsterdam: 1977).  ISBN 0-444-86388-5 pbk.
 
[Barendregt1981]
Barendregt, Hendrik Pieter. The Lambda Calculus: Its Syntax and Semantics. North-Holland (Amsterdam, 1981). ISBN 0-444-85490-8.  Studies in Logic and the Foundations of Mathematics, vol. 103.
     This is one of my fundamental sources on the lambda calculus.  I want to pay particular attention to combinatory logic (CL) and combinatory algebra (CA).  My notes on this book focus on that.
   Content
     Preface
     Part I. Towards the Theory
          1. Introduction
          2. Conversion
          3. Reduction
          4. Theories
          5. Models
     Part II. Conversion
          6. Classical Lambda Calculus
          7. The Theory of Combinators
          8. Classical Lambda Calculus (continued)
          9. The λI-Calculus
          10. Böhm Trees
     Part III. Reduction
          11. Fundamental Theorems
          12. Strongly Equivalent Reductions
          13. Reduction Strategies
          14. Labelled Reduction
          15. Other Notions of Reduction
     Part IV. Theories
          16. Sensible Theories
          17. Other Lambda Theories
     Part V. Models
          18. Construction of Models
          19. Local Structure of Models
          20. Global Structure of Models
          21. Combinatory Groups
     Appendices
          Appendix A. Typed Lambda Calculus
          Appendix B. Illative Combinatory Logic
          Appendix C. Variables
     References
     Index of Names
     Index of Definitions
     Index of Symbols
 
[Barwise1977]
Barwise, Jon (ed.).  Handbook of Mathematical Logic.  Elsevier Science BV (Amsterdam: 1977).  ISBN 0-444-86388-5 pbk.
     I have misgivings about this book.  When I ordered it, I didn't realize that it was the same book I had owned once before.  Part of the problem is that it seems like there is something missing.   I get the feeling that much of this material only makes sense to those who wrote it (something I have to watch out for also).  Along with that, I dread that there are errors in here that I won't be able to figure out.  I just get the feeling that I am dealing with something careless and that I won't be able to master it.  Even so, I am keeping this one around in anticipation that doors will open and that I will find that the book delivers on its promise.  dh: 2000-11-23
     "The Handbook of Mathematical Logic is an attempt to share with the entire mathematical community some modern developments in logic.  We have selected from the wealth of topics available some of those which deal with the basic concepts of the subject, or are particularly important for applications to other parts of mathematics, or both.
     "Mathematical logic is traditionally divided into four parts: model theory, set theory, recursion theory and proof theory.  We have followed this division ... .  The first chapter or two in each part are introductory in scope.  More advanced chapters follow, as do chapters on applied or applicable parts of mathematical logic.  Each chapter is definitely written for someone who is not a specialist in the field in question. ... 
     "We hope that many mathematicians will pick up this book out of idle curiosity and leaf through it to get a feeling for what is going on in another part of mathematics.  It is hard to imagine a mathematician who could spend ten minutes doing this without wanting to pursue a few chapters, and the introductory sections of others, in some detail.  It is an opportunity that hadn't existed before and is the reason for the Handbook."  -- From the Foreword, p. vii.
  Contents
     Foreword
     Contributors
     Part A: Model Theory
          Guide to Part A
          A.1. An introduction to first-order logic, Jon Barwise
          A.2. Fundamentals of model theory, H. Jerome Keisler
          A.3. Ultraproducts for algebraists, Paul C. Eklof
          A.4. Model completeness, Angus Macintyre
          A.5. Homogenous sets, Michael Morley
          A.6. Infinitesimal analysis of curves and surfaces, K. D. Stroyan
          A.7. Admissible sets and infinitary logic, M. Makkai
          A.8. Doctrines in categorical logic, A. Kock and G. E. Reyes
     Part B: Set Theory
          Guide to Part B
          B.1. Axioms of set theory, J. R. Shoenfield
          B.2. About the axiom of choice, Thomas J. Jech
          B.3. Combinatorics, Kenneth Kunen
          B.4. Forcing, John P. Burgess
          B.5. Constructibility, Keith J. Devlin
          B.6. Martin's Axiom, Mary Ellen Rudin
          B.7. Consistency results in topology, I. Juhász
     Part C: Recursion Theory
          Guide to Part C
          C.1. Elements of recursion theory, Herbert B. Enderton
          C.2. Unsolvable problems, Martin Davis
          C.3. Decidable theories, Michael O. Rabin
          C.4. Degrees of unsolvability: a survey of results, Stephen G. Simpson
          C.5. a-recursion theory, Richard A. Shore
          C.6. Recursion in higher types, Alexander Kechris and Yiannis N. Moschovakis
          C.7. An introduction to inductive definitions, Peter Aczel
          C.8. Descriptive set theory: Projective sets, Donald A. Martin
     Part D: Proof Theory and Constructive Mathematics
          Guide to Part D
          D.1. The incompleteness theorems, C. Smorynski
          D.2. Proof theory: Some applications of cut-elimination, Helmut Schwichtenberg
          D.3. Herbrand's Theorem and Gentzen's notion of a direct proof, Richard Statman
          D.4. Theories of finite type related to mathematical practice, Solomon Feferman
          D.5. Aspects of constructive mathematics, A. S. Troelstra
          D.6. The logic of topoi, Michael P. Fourman
          D.7. The type free lambda calculus, Henk Barendregt
          D.8. A mathematical incompleteness in Peano Arithmetic, Jeff Paris and Leo Harrington
     Author Index
     Subject Index
   
[Bernays1968]
Bernays, Paul.  Axiomatic Set Theory.  With a historical introduction by Abraham A. Fraenkel. 2nd edition.  Studies in Logic and The Foundations of Mathematics.  North-Holland (Amsterdam: 1958, 1968).  Unabridged and unaltered republication by Dover Publications (New York: 1991).  ISBN 0-486-66637-9 pbk.
     The first part of this text, by Abraham Fraenkel, is valuable as a survey of the effort to deal with the inconsistencies that lurked in Cantor's formulation.  Fraenkel presents Zermelo's system in that light, and provides his own modification.  The approaches of Russell, Quine, von Neumann and Bernays are contrasted, among others.  The motivation for a complete axiomatization is to establish the (likely) consistency of the theory, with due respect to Gödel, and to also find ways to embrace as much as possible without crossing the line into inconsistency.  
     Bernays, for his part, provides a detailed axiomatic formulation and traces the origin of his axioms and their consequences and related definitions.   The progression is increasingly abstract, with historical connections accounted for.
     Since ZF (or ZFC) seems destined to stick around, a development of it in more-contemporary language is given by [Suppes1972].  Quine draws further contrast, with more details of von Neumann's approach, while contrasting his own efforts in [Quine1969].  Bernays does not keep the emphasis that von Neumann gave to functions, and that approach, of potential value in computational contexts, must be found elsewhere. -- dh:2002-07-24
   Content
     Preface
     Part I. Historical Introduction
          1. Introductory Remarks
          2. Zermelo's System.  Equality and Extensionality
          3. "Constructive" Axioms of "General" Set Theory
          4. The Axiom of Choice
          5. Axioms of Infinity and of Restriction
          6. Development of Set-Theory from the Axioms of Z
          7. Remarks on the Axiom Systems of von Neumann, Bernays, Gödel
     Part II. Axiomatic Set Theory
          Introduction
          Chapter I.  The Frame of Logic and Class Theory
          Chapter II. The Start of General Set Theory
          Chapter III.  Ordinals; Natural Numbers; Finite Sets
          Chapter IV. Transfinite Recursion
          Chapter V. Power; Order; Wellorder
          Chapter VI.  The Completing Axioms
          Chapter VII.  Analysis; Cardinal Arithmetic; Abstract Theories
          Chapter VIII. Further Strengthening of the Axiom System
     Index of Authors (Part I)
     Index of Symbols (Part II)
     Index of Matters (Part II)
     List of Axioms (Part II)
     Bibliography (Part I and II)
   
[Boole1854]
Boole, George.  An Investigation of the Laws of Thought on which Are Founded the Mathematical Theories of Logic and Probabilities.  Macmillan (Toronto, London: 1854).  Dover edition (New York: 1958) with all corrections made in the text.  ISBN 0-486-60028-9.
   Contents
     Preface
     I. Nature and Design of this Work
     II. Signs and their Laws
     III. Derivation of the Laws
     IV. Division of Propositions
     V. Principles of Symbolic Reasoning
     VI. Of Interpretation
     VII. Of Elimination
     VIII. Of Reduction
     IX. Methods of Abbreviation
     X. Conditions of a Perfect Method
     XI. Of Secondary Propositions
     XII. Methods in Secondary Propositions
     XIII. Clarke and Spinoza
     XIV. Example of Analysis
     XV. Of the Aristotelian Logic
     XVI. Of the Theory of Probabilities
     XVII. General Method in Probabilities
     XVIII. Elementary Illustrations
     XIX. Of Statistical Conditions
     XX. Problems on Causes
     XXI. Probability of Judgments
     XXII. Constitution of the Intellect 
    
[Boolos1993b]
Boolos, George.  The Logic of Provability.  Cambridge University Press (Cambridge: 1993).  ISBN 0-521-48325-5 pbk.
     "When modal logic is applied to the study of provability, it becomes provability logic.  This book is an essay on provability logic."  -- from the Preface, p. ix.
   Content
     Preface
     Introduction
     1. GL and Other Systems of Propositional Logic
     2. Peano Arithmetic
     3. The box as Bew(x)
     4. Semantics for GL and other Modal Logics
     5. Completeness and Decidability of GL and K, K4, T, B, S4, and S5
     6. Canonical Models
     7. On GL
     8. The Fixed Point Theorem
     9. The Arithmetical Completeness Theorems for GL and GLS
     10. Trees for GL
     11. An Incomplete System of Modal Logic
     12. An S4-Preserving Proof-Theoretical Treatment of Modality
     13. Modal Logic within Set Theory
     14. Modal Logic within Analysis
     15. The Joint Provability Logic of Consistency and ω-Consistency
     16. On GLB: The Fixed Point Theorem, Letterless Sentences, and Analysis
     17. Quantified Provability Logic
     18. Quantified Provability Logic with One One-Place Predicate Letter
     Notes
     Bibliography
     Index
     Notation and Symbols
   
[Boolos1998]
Boolos, George.  Burgess, John P., Jeffrey, Richard (ed.).  Logic, Logic, and Logic.  Harvard University Press (Cambridge, MA: 1998).  With Introductions and Afterword by John P. Burgess.  ISBN 0-674-53767-X pbk.
     This collection of articles by George Boolos provides the more accessible works not part of the work on provability theory and not strenuously technical.  There are a number of skeptical accounts on set theory and logic that are useful in understanding pitfalls that abide in formulations such as ZFC.  Whether this provides sufficient cause for caution in ones reliance on the accepted applications of logic and set theory, one will have to divine by examining the topics here more closely.  -- dh:2002-07-26
   Content
     Editorial Preface (John P. Burgess and Richard Jeffrey)
     Editor's Acknowledgments
     I. Studies on Set Theory and the Nature of Logic
          Introduction
          1. The Interative Concept of Set [1971]
          2. Reply to Charles Parsons' "Sets and Classes" [1974b, first publication here]
          3. On Second-Order Logic [1975c]
          4. To Be is to Be a Value of a Variable (or to Be Some Values of Some Variables) [1984e]
          5. Nominalist Platonism [1985c]
          6. Iteration Again [1989a]
          7. Introductory Note to Kurt Gödel's "Some Basic Theorems on the Foundations of Mathematics and their Implications" [1995b]
          8. Must We Believe in Set Theory [1997d]
     II. Frege Studies
          Introduction
          9. Gottlob Frege and the Foundations of Arithmetic 11997b, first publication here]
          10. Reading the Begriffsschrift [1985d]
          11. Saving Frege from Contradiction [1986-87]
          12. The Consistency of Frege's Foundations of Arithmetic [1987a]
          13. The Standard of Equality of Numbers [1990e]
          14. Whence the Contradiction [1993]
          15. 1879? [1994a]
          16. The Advantages of Honest Toil over Theft [1994b]
          17. On the Proof of Frege's Theorem [1996b]
          18. Frege's Theorem and the Peano Postulates [1995a]
          19. Is Hume's Principle Analytic? [1997c]
          20. Die Grundlagen der Arithmetik, §§82-83 (with Richard Heck) [1997]
          21. Constructing Cantorian Counterexamples (with a note by Vann McGee) [1997a]
     III. Various Logical Studies and Lighter Papers
          Introduction
          22. Zooming Down the Slippery Slope [1991]
          23. Don't Eliminate Cut [1984a]
          24. The Justification of Mathematical Induction [1985b]
          25. A Curious Inference [1987b]
          26. A New Proof of the Gödel Incompleteness Theorem [1989b]
          27. On "Seeing" the Truth of the Gödel Sentence [1990b]
          28. Quotational Ambiguity [1995c]
          29. The Hardest Logical Puzzle Ever [1996a]
          30. Gödel's Second Incompleteness Theorem Explained in Words of One Syllable [1994c]
     Afterword
     Bibliography
     Index
   
[Boolos2002]
Boolos, George S., Burgess, John P., Jeffrey, Richard.  Computability and Logic. ed.4.  Cambridge University Press (Cambridge: 2002).  ISBN 0-521-00758-5 pbk.
     The three previous editions, by Boolos and Jeffrey, were published in 1974, 1985, and 1989.  This is one of those books that is restless on my bookshelf.  Is it logic?  Computability?  No logic.  Like that.  I collect my notes here, under Logic, because it is a fitting companion to [Boolos1993b] and [Boolos1998].  More than that, from the very outset the book takes a logical stance, addressing some of the key questions of set theory that impinge on computation and effective computability.  The result is very economical in coming at the key questions around computation and the relationship of computation and logic. -- dh:2002-09-07
   Content
     Preface
     Computability Theory
     1.  Enumerability
     2.  Diagonalization
     3.  Turing Computability
     4.  Uncomputability
     5.  Abacus Computability
     6.  Recursive Functions
     7.  Recursive Sets and Relations
     8.  Equivalent Definitions of Computability
     Basic Metalogic
     9.  A Précis of First-Order Logic: Syntax
     10. A Précis of First-Order Logic: Semantics
     11. The Undecidability of First-Order Logic
     12. Models
     13. The Existence of Models
     14. Proofs and Completeness
     15. Arithmetization
     16. Representability of Recursive Functions
     17. Indefinability, Undecidability, Incompleteness
     18. The Unprovability of Consistency
     Further Topics
     19. Normal Forms
     20. The Craig Interpolation Theorem
     21. Monadic and Dyadic Logic
     22. Second-Order Logic
     23. Arithmetic Definability
     24. Decidability of Arithmetic without Multiplication
     25. Nonstandard Models
     26. Ramsey's Theorem
     27. Modal Logic and Provability
     Hints for Selected Problems
     Annotated Bibliography
     Index
   
[Burgess1998]
Burgess, John P.  Introductions and Afterword in [Boolos1998]
   
[Burgess2002]
Boolos, George S., Burgess, John P., Jeffrey, Richard.  Computability and Logic. ed.4.  Cambridge University Press (Cambridge: 2002).  ISBN 0-521-00758-5 pbk.  See [Boolos2002]
    
[Cantor1915]
Cantor, Georg.  Contributions to the Founding of the Theory of Transfinite Numbers.  Translation, Introduction and Notes by Philip E.  B. Jourdain.  Open Court (London: 1915). Unabridged and unaltered republication by Dover Publications (New York: 1955).  ISBN 0-486-60045-9 pbk.
     "Wierstrass [starting in the 1840's] carried research into the principles of arithmetic farther than it had been carried before.  But we must also realize that there were questions, such as the nature of the whole number itself, to which he made no valuable contributions.  These questions, though logically the first in arithmetic, were, of course, historically the last to be dealt with.  Before this could happen, arithmetic had to receive a development by means of Cantor's discovery of transfinite numbers, into a theory of cardinal and ordinal numbers, both finite and transfinite, and logic had to be sharpened, as it was by Dedekind, Frege, Peano and Russell--to a great extent owing to the needs which this theory made evident."  From the Introduction, p.23.
     "In 1873, Cantor set out from the question whether the linear continuum (of real numbers) could be put in a one-one correspondence with the aggregate of the whole numbers, and found the rigorous proof that this is not the case.  This proof ... was published in 1874."  From the Introduction, p.38.
     "... Conception of  (1, 1)-correspondence between aggregates was the fundamental idea in a memoir of 1877, published in 1878, in which some important theorems of this kind of relation between various aggregates were given and suggestions made of a classification of aggregates on this basis.
     "If two well-defined aggregates can be put into such a (1, 1)-correspondence (that is to say, if, element to element, they can be made to correspond completely and uniquely), they are said to be of the same "power" (Mächtigkeit) or to be "equivalent" (aequivalent).  When an aggregate is finite, the notion of power corresponds to that of number (Anzahl), for two such aggregates have the same power when, and only when, the number of their elements is the same.
     "A part (Bestandteil; any other aggregate whose elements are also elements of the original one) of a finite aggregate has always a power less than that of the aggregate itself, but this is not always the case with infinite aggregates--for example, the series of positive integers is easily seen to have the same power as that part of it consisting of the even integers--and hence, from the circumstances that an infinite aggregate M is part of N (or is equivalent to a part of N), we can only conclude that the power of M is less than that of N if we know that these powers are unequal."  From the Introduction, pp.40-41.  
     Cantor (p.86) defines "part" as Jordain does, assuming that Jordain's translation is faithful.  It is what we now call a proper subset.
     Over time, Cantor also addressed the conditions for an aggregate being well-defined and being enumerable--equivalent to the set of natural numbers, and this is laid out by Jordain as preparation for the Begründung translated in this work. 
     In the key papers translated here, Cantor summarizes the conception of a set (aggregate: Menge) in four pages (85-89).  The work moves on through the development of transfinite cardinals to applicability of this powerful instrument to analysis.  This is Jourdain's justification for changing the title [Preface, p.v].  Reading it today, I would say that Cantor is not directly restricting himself to "numbers" even though he may well have that application in mind and, in some sense, numbers can't be escaped.  (I suppose Pythagoras would be dumbfounded as well as pleased.)  It seems to me that Cantor knew exactly what he was doing and the original title should stand.  Hence my classification of the work under logic (including set theory).   dh:2002-06-18.
   Contents
     Preface [Jourdain 1915]
     Table of Contents
     Introduction [Jourdain 1915]
     Contributions to the Founding of the Theory of Transfinite Numbers [Beiträge zur Begründung der transfiniten Mengenlehre]
          Article I (1895)
          Article II (1897)
     Notes [Jourdain 1915]
     Index
   
[Church1936]
Church, Alonzo.  An Unsolvable Problem of Elementary Number Theory.  American Journal of Mathematics 58 (1936), 345-363.  Reprinted in pp. 88-115 with supplemental notes in [Davis1965]
     This is the paper in which Church makes the assertion since known as Church's Thesis (and lately, the Church-Turing Thesis), in the last sentence, and its footnote, in section 1.  Subsequent sections refer to work to appear or underway related to investigations by Church, Rosser, Kleene, Gödel, and others.  .
   Content
     1. Introduction
     2. Conversion and λ-definability
     3. The Gödel representation of a formula
     4. Recursive functions
     5. Recursiveness of the Kleene p-function
     6. Recursiveness of certain functions of formulas
     7. The notion of effective calculability
     8. Invariants of conversion
   
[Church1940]
Church, Alonzo.  A formulation of the Simple Theory of Types.  J. Symbolc Logic 5, 2 (June 1940), 56-48.  DOI: 10.2307/22661700 available on JSTOR <https://www.jstor.org/stable/226170>.
   Identified as inspirational in [Scott1993] and [Paulson2018], this work identifies simple type ο (omicron) as the type of propositions and represents a logic thereby. For a lower-level computational model such as for oMiser, there need not be an explicit type; discriminators such as Uβ accomplish this without a distinct type, just as for early programming languages, including LISP.  The notation for functional type has also been refined over the years, achieving consistent widespread usage.
  
[Church1941]
Church, Alonzo.  The Calculi of Lambda-Conversion.  Annals of Mathematics Studies 6.  Princeton University Press (Princeton: 1941).  1951 Second printing with bibliography addenda ISBN 0-691-08394-0 pbk.
  Content
     I. Introduction
        1. The concept of a function
        2. Extension and intension
        3. Functions of several variables
        4. Abstraction
     II. Lambda-Conversion
        5. Primitive symbols, and formulas
        6. Conversion
        7. Fundamental theorems on well-formed formulas and on the normal form
     III. Lambda-Definability
        8. Lambda-definability of functions of positive integers
        9. Ordered pairs and traids, the predecessor function
       10. Propositional functions, the Kleene p-function
       11. Definition by recursion
      IV. Combinations, Gödel Numbers
       12. Combinations
       13. Primitive sets of formulas
       14. An application of the theory of combinations
       15. A combinatory equivalent of conversion
       16. Gödel numbers
      V. The Calculi of λ-K-Conversion and λ-δ-Conversion
        17. The calculus of λ-K-conversion
        18. The calculus of restricted λ-K-conversion
        19. Transfinite ordinals
        20. The calculus of λ-δ-conversion
        21. A system of symbolic logic
      Index of the Principal Formulas Introduced by Definition
      Bibliography
      Corrections and Additions
  
[Church1956]
Church, Alonzo.  Introduction to Mathematical Logic.  Princeton University Press (Princeton, NJ: 1944, 1956).  ISBN 0-691-02906-7 pbk.  With 1958 errata.
     Originally identified as "Volume I," the material has been expanded and updated, and Volume II is destined to never appear, at this point.  I give the expansion of the Introduction content to identify areas that students may want to explore in understanding Church's approach to mathematical logic.  
     "In order to set up a formalized language we must of course make use of a language already known to us, say English or some portion of the English language, stating in that language the vocabulary and rules of the formalized language.  This procedure is analogous to that familiar to the reader in language study--as, e.g., in the use of a Latin grammar written in English--but differs in the precision with which rules are stated, in the avoidance of irregularities and exceptions, and in the leading idea that the rules of the language embody a theory or system of logical analysis.
     "The device of employing one language in order to talk about another is one for which we shall have frequent occasion not only in setting up formalized languages but also in making theoretical statements as to what can be done in a formalized language, our interest in formalized languages being less often in their actual and practical use as languages than in the general theory of such use and in its possibilities in principle.  Whenever we employ a language in order to talk about some other language (itself or another), we shall call the latter language the object language, and we shall call the former the meta-language." -- From section 07, The logistic method, p.47.
  Content
     Preface
     Introduction
          00. Logic
          01. Names
          02. Constants and variables
          03. Functions
          04. Propositions and propositional functions
          05. Improper symbols, connectives
          06. Operators, quantifiers
          07. The logistic method
          08. Syntax
          09. Semantics
     I. The Propositional Calculus
     II. The Propositional Calculus (Continued)
     III. Functional Calculi of First Order
     IV. The Pure Functional Calculi of First Order
     V. Functional Calculi of Second Order
     Index of Definitions
     Index of Authors Cited
     Errata
   
[CL2018]
Combinatory Logic.  Wikipedia Article.  Accessed on 2018-05-15. Re-accessed on 2024-07-25.
   2024-07-25: I find that Wikipedia articles on abstruse topics tend to be hyper-abstruse to the next level.  I'm also unclear how that can be resolved within the guidelines for Wikipedia articles.  I do recall there being near-tutorial articles in encyclopedias during my high-school fascination with how one could assay the chemical composition of minerals.  Perhaps I was naive. 
   One might also wonder what accounts for the apparent instability of articles of this kind when the History is viewed.  The article's Talk page may be useful in some respects, although it is not always obvious how the article was influenced, if at all.  Now I wonder how it came about that I understood the introduction in [Rosenbloom1950] without so much difficulty. 
    I cite this article mainly for its discussion and some ideas, with a caution that if something doesn't seem to make sense it might be good to find a different source for comparison; the historical discussion and mentions of resources are still somewhat usable.  Important questions related to mathematical logic are addressed here, although understanding that seems inessential, as much as it does indicate what the interest of theoreticians is all about.
   For an intense treatment, consider [Barendregt1981: Chapter 7].  My own introduction was [Rosenbloom1950: Section 4, starting on p.111].  [Burge1975: Section 1.9] is perhaps the most straightforward with regard to functional-/applicative-programming employment, in contrast/conjunction with Lambda Conversion (Section 1.8 there).  The coverage of [Barendregt1977] is a valuable background and guide into the formal mathematical-logic considerations.
   The origin of combinatory logic is available [Schönfinkel1924].  The introduction of [Quine1977] is valuable, along with understanding of the motivation for eliminating quantifiers and quantified terms.
  
[Copi1978]
Copi, Irving M.  Introduction to Logic,  ed. 5.  Macmillan (New York: 1953, 1961, 1968, 1972, 1978).  ISBN 0-02-324880-7.
     "There are obvious benefits to be gained from the study of logic: heightened ability to express ideas clearly and concisely, increased skill in defining one's terms, enlarged capacity to formulate arguments rigorously and to analyze them critically.  But the greatest benefit, in my judgment, is the recognition that reason can be applied in every aspect of human affairs."  Preface, p.vii.
   Content
     Preface
     Part One: Language
          1. Introduction
          2. The Uses of Language
          3. Informal Fallacies
          4. Definition
     Part Two: Deduction
          5. Categorical Propositions
          6. Categorical Syllogisms
          7. Arguments in Ordinary Language
          8. Symbolic Logic
          9. The Method of Deduction
          10. Quantification Theory
     Part Three: Induction
          11. Analogy and Probable Inference
          12. Causal Connections: Mill's Methods of Experimental Inquiry
          13. Science and Hypothesis
          14. Probability
     Solutions to Selected Exercises
     Special Symbols
     Index
   
[Curry1977]
Curry, Haskell B.  Foundations of Mathematical Logic.  Dover Publications (New York: 1963, 1977).  ISBN 0-486-63462-0 pbk.
     "... This book is intended to be self-contained.  It aims to give a thorough account of a part of mathematical logic which is truly fundamental, not in a theoretical or philosophical sense, but from the standpoint of a student; a part which needs to be thoroughly understood, not only by those who will later become specialists in logic, but by all mathematicians, philosophers, and scientists whose work impinges upon logic.
     "The part of mathematical logic which is selected for treatment may be described as the constructive theory of the first-order predicate calculus.  That this calculus is central in modern mathematical logic does not need to be argued.  Likewise, the constructive aspects of this calculus are fundamental for its higher study.  Furthermore, it is becoming increasingly apparent that mathematicians in general need to be aware of the difference between the constructive and the nonconstructive, and there is hardly any better way of increasing this awareness than by giving a separate treatment of the former.  Thus there seems to be a need for a graduate-level exposition of this fundamental domain."  -- From the Preface, p.iii.
   Content
     Preface to the Dover Edition
     Preface
     Explanation of Conventions
     1. Introduction
          A. The nature of mathematical logic
          B. The logical antinomies
          C. The nature of mathematics
          D. Mathematics and logic
          S. Supplementary topics
     2. Formal Systems
          A. Preliminaries
          B. Theories
          C. Systems
          D. Special forms of systems
          E. Algorithms
          S. Supplementary topics
     3. Epitheory
          A. The nature of epitheory
          B. Replacement and monotone relations
          C. The theory of definition
          D. Variables
          S. Supplementary topics
     4. Relational Logical Algebra
          A. Logical algebras in general
          B. Lattices
          C. Skolem lattices
          D. Classical Skolem lattices
          S. Supplementary topics
     5. The Theory of Implication
          A. General principles of assertional logical algebra
          B. Propositional algebras
          C. The systems LA and LC
          D. Equivalence of the systems
          E. L deducibility
          S. Supplementary topics
     6. Negation
          A. The nature of negation
          B. L systems for negation
          C. Other formulations of negation
          D. Technique of classical negation
          S. Supplementary topics
     7. Quantification
          A. Formulation
          B. Theory of the L* systems
          C. Other forms of quantification theory
          D. Classical epitheory
          S. Supplementary topics
     8. Modality
          A. Formulation of necessity
          B. The L theory of necessity
          C. The T and H formulations of necessity
          D. Supplementary topics
     Bibliography
     Index
   
[Davis1965] 
Davis, Martin (ed.). The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions. Raven Press (New York: 1965). ISBN 0-911216-01-4. 
   Content
     Kurt Gödel
          On Formally Undecidable Propositions of the Principia Mathematica and Related Systems, I [1931, translated from the German by Elliott Mendelson]
          On undecidable propositions of formal mathematical systems [1934 notes by S. C. Kleene and J. B. Rosser with 1964 postscriptum by Gödel ]
          On Intuitionistic Arithmetic and Number Theory [1933e, translated from the German by Martin Davis]
          On the Length of Proofs [1936a, translated from the German by Martin Davis]
          Remarks Before the Princeton Bicentennial Conference on Problems in Mathematics [1946]
     Alonzo Church
          An unsolvable problem of elementary number theory [1936]
          A Note on the Entscheidungsproblem [1936a]
     Alan M. Turing
          On computable numbers, with an application to the entscheidungsproblem [1936 with 1937 corrections]
          Systems of Logic Based on Ordinals [1939]
     J. B. Rosser
          An Informal Exposition of Proofs of Gödel's Theorem and Church's Theorem [1939]
          Extensions of Some Theorems of Gödel and Church [1936]
     Stephen C. Kleene
          General Recursive Functions of Natural Numbers [1936 with 1938 corrections]
          Recursive Predicates and Quantifiers [1943]
     Emil Post
          Finite Combinatory Processes, Formulation I [1936]
          Recursive Unsolvability of a Problem of Thue [1947]
          Recursively enumerable sets of positive integers and their decision problems [1944]
          Absolutely Unsolvable Problems and Relatively Undecidable Propositions -- Account of an Anticipation [1941 published 1964]
     Index
   
[Davis2000]
Davis, Martin.  Engines of Logic: Mathematicians and the Origin of the Computer.   W. W. Norton (New York: 2000).  ISBN 0-393-32229-7 pbk.  Paperback edition of book originally published as The Universal Computer: The Road from Liebniz to Turing.
     "As computers have evolved from the room-filling behemoths that were the computers of the 1950s to the small, powerful machines of today that perform a bewildering variety of tasks, their underlying logic has remained the same.  These logical concepts have developed out of the work of a number of gifted thinkers over a period of centuries.  In this book I tell the story of the lives of these people and explain some of their thoughts.  The stories are fascinating in themselves, and my hope is that readers will not only enjoy them, but that they will also come away with a better sense of what goes on insider their computers and with an enhanced respect for the value of abstract thought."  -- from the Preface, p. ix.
     I really do discipline myself -- sometimes successfully -- to leave books on their shelves, a practice best sustained by avoiding bookstores.  The other day, while shopping for a specific book and thereby vulnerable, my thoughts were on "the unusual effectiveness of mathematics." I opened Engines of Logic to see what Davis has to say about Einstein saying anything.  Although I found no direct connection on the peculiar-seeming harmony of theory and reality in the 8 places (and further in the notes) where Einstein figures in this dance, I was led to the discussion of Hilbert's life and the important meetings in Königsberg (pp. 102-105).  I was startled to see the connections among the players in modern logic, and also be reminded of the terrible events of and between the World Wars and how this led to the great dispersal in which Princeton's Institute for Advanced Study arose as a safe haven.  Reading Hilbert's epitaph, I wept silently as I walked to the checkout counter.  
     Despite repeated evidence, I am regularly surprised by the personal aspects of the lives of mathematicians and those singular individuals who have forever altered our view of the world and demonstrated the power of abstract conceptions in their immortal legacy.  There is an amazing, connected community of participants, colleagues, adversaries, teachers, students and scholars who knew each other as correspondents, as professors, and as compatriots and friends in a chain of lived relationships on which was anchored the development of mathematical logic and the practical creation of the computer.  This book brings the humanity of the mathematician's world to life for me.   I recommend it along with the many sources in its notes and references.  It evokes for me the same passion that I awaken on re-reading Berlinski's books (on calculus and on the algorithm) and  the venerable Men of Mathematics. -- dh:2002-09-05
   Content
     Preface
         Note to the Paperback Edition
     Introduction
     1.  Liebniz's Dream
     2.  Boole Turns Logic into Algebra
     3.  Frege: From Breakthrough to Despair
     4.  Cantor: Detour through Infinity
     5.  Hilbert to the Rescue
     6.  Gödel Upsets the Applecart
     7.  Turing Conceives of the All-Purpose Computer
     8.  Making the First Universal Computers
     9.  Beyond Liebniz's Dream
     Epilogue
     Notes
     References
     Index
   
[Dawson1986]
Gödel, Kurt., Feferman, Solomon (editor-in-chief), Dawson, John W. Jr., Kleene, Stephen C., Moore, Gregory H., Solovay, Robert M., van Heijenoort, Jean (eds.).  Kurt Gödel: Collected Works, vol.1: Publications 1929-1936.  Oxford University Press (New York: 1986).  ISBN 0-19-514720-0 pbk.  See [Gödel1986]
   
[Dawson1990]
Gödel, Kurt., Feferman, Solomon (editor-in-chief), Dawson, John W. Jr., Kleene, Stephen C., Moore, Gregory H., Solovay, Robert M., van Heijenoort, Jean (eds.).  Kurt Gödel: Collected Works, vol.2: Publications 1938-1974.  Oxford University Press (New York: 1990).  ISBN 0-19-514721-9 pbk.  See [Gödel1990]
   
[Dawson1995]
Gödel, Kurt., Feferman, Solomon (editor-in-chief)., Dawson, John W. Jr., Goldfarb, Warren., Parsons, Charles., Solovay, Robert M. (eds.).  Kurt Gödel: Collected Works, vol.3: Unpublished essays and lectures.  Oxford University Press (New York: 1995).  ISBN 0-19-514722-7 pbk.  See [Gödel1995]
   
[Enderton2001]
Enderton, Herbert B.  A Mathematical Introduction to Logic.  ed.2.  Harcourt/Academic Press (Burlington, MA: 1972, 2001).  ISBN 0-12-238452-0.
     "The book is intended for the reader who has not studied logic previously, but who has some experience in mathematical reasoning.  There are no specific prerequisites aside from a willingness to function at a certain level of abstraction and rigor.  There is the inevitable use of basic set theory.  Chapter 0 gives a concise summary of the set theory used.  One should not begin the book by studying this chapter; it is instead intended for reference if and when the need arises." -- from the Preface, p.x.
     When asked what I recommend to computer scientists for delving deeper into logic and its connections with computation and language, I recommend two books.  Stolyar's elementary text as a starter, with Enderton's book as more-comprehensive but still-accessible introduction to further concepts that arise in the application of logic to mathematical subjects, including computer science.  Enderton provides a coherent progression through topics that I have encountered only by happenstance in earlier forays.  There is appropriate rigor and an useful foundation that I will certainly appropriate in my work and in discussions with others.  This book is a great place to sharpen ones understanding and application of logic and also as a place to direct others as a basis for a common background in theoretical explorations  -- dh:2002-07-16.
   Content
     Preface
     Introduction
     Chapter Zero.  Useful Facts About Sets
     Chapter One.  Sentential Logic
          1.0 Informal Remarks on Formal Languages
          1.1 The Language of Sentential Logic
          1.2 Truth Assignments
          1.3 A Parsing Algorithm
          1.4 Induction and Recursion
          1.5 Sentential Connectives
          1.6 Switching Circuits
          1.7 Compactness and Effectiveness
     Chapter Two. First-Order Logic
          2.0 Preliminary Remarks
          2.1 First-Order Languages
          2.2 Truth and Models
          2.3 A Parsing Algorithm
          2.4 A Deductive Calculus
          2.5 Soundness and Completeness Theorems
          2.6 Models of Theories
          2.7 Interpretations Between Theories
          2.8 Nonstandard Analysis
     Chapter Three.  Undecidability
          3.0 Number Theory
          3.1 Natural Numbers with Successor
          3.2 Other Reducts of Number Theory
          3.3 A Subtheory of Number Theory
          3.4 Arithmetization of Syntax
          3.5 Incompleteness and Undecidability
          3.6 Recursive Functions
          3.7 Sound Incompleteness Theorem
          3.8 Representing Exponentiation
     Chapter Four.  Second-Order Logic
          4.0 Second-Order Languages
          4.1 Skolem Functions
          4.2 Many-Sorted Logic
          4.3 General Structures
     Suggestions for Further Reading
     List of Symbols
     Index
   
[Feferman1977]
Feferman, Solomon.  Theories of Finite Type Related to Mathematical Practice.  Chapter D.4 in The Handbook of Mathematical Logic, Jon Barwise, editor.  Elsevier (Amsterdam: 1977).  ISBN 0-444-86388-5 pbk.
  
[Feferman1986]
Gödel, Kurt., Feferman, Solomon (editor-in-chief), Dawson, John W. Jr., Kleene, Stephen C., Moore, Gregory H., Solovay, Robert M., van Heijenoort, Jean (eds.).  Kurt Gödel: Collected Works, vol.1: Publications 1929-1936.  Oxford University Press (New York: 1986).  ISBN 0-19-514720-0 pbk.  See [Gödel1986]
   
[Feferman1990]
Gödel, Kurt., Feferman, Solomon (editor-in-chief), Dawson, John W. Jr., Kleene, Stephen C., Moore, Gregory H., Solovay, Robert M., van Heijenoort, Jean (eds.).  Kurt Gödel: Collected Works, vol.2: Publications 1938-1974.  Oxford University Press (New York: 1990).  ISBN 0-19-514721-9 pbk.  See [Gödel1990]
   
[Feferman1995]
Gödel, Kurt., Feferman, Solomon (editor-in-chief)., Dawson, John W. Jr., Goldfarb, Warren., Parsons, Charles., Solovay, Robert M. (eds.).  Kurt Gödel: Collected Works, vol.3: Unpublished essays and lectures.  Oxford University Press (New York: 1995).  ISBN 0-19-514722-7 pbk.  See [Gödel1995]
   
[Forster1995]
Forster, T. E.  Set Theory with a Universal Set: Exploring an Untyped Universe.  ed.2.  Oxford University Press (Oxford: 1992, 1995).  ISBN 0-19-851477-8.
     "NF is a much richer and more mysterious system than the other set theories with a universal set, and there are large areas in its study (e.g., the reduction of the consistency question) which have no counterparts elsewhere in the study of set theories with V ∈ V.  There just is a great deal more to say about NF than about the other systems."  Preface to the First Edition, p.v.
     [dh:2004-02-19] I was led here by some startling references to Quine's New Foundations (NF) on some discussion lists that I follow.  This is often in the context of "ur-elements" and also "avoiding problems" or "applicable in computational models."  It seemed wise to find out what that is about.  I learned that there is an active community of interest in NF, and that Thomas Forster's work is prized as a valuable current treatment.  I am counting on the first two sections to provide equipment for comprehending these mentions.  But first, I must equip myself to comprehend the rather technical first two sections.  Meanwhile, I can simply enjoy the way Forster writes while I read for the gist of it.
   Content
     Preface to the First Edition
     Preface to the Second Edition
     1. Introduction
          1.1 Annotated definitions
          1.2 Some motivations and axioms
          1.3 A brief survey
          1.4 How do theories with V ∈ V avoid the paradoxes?
          1.5 Chronology
     2. NF and Related Systems
          2.1 NF
          2.2 Cardinal and ordinal arithmetic
          2.3 The Kaye-Specker equiconsistency lemma
          2.4 Subsystems, term models, and prefix classes
          2.5 The converse consistency problem
     3. Permutation Models
     4. Church-Oswald Models
     5. Open Problems
     Bibliography
     Index of Definitions
     Author Index
     General Index
   
[Forster2003]
Forster, Thomas.  Reasoning About Theoretical Entities.  Advances in Logic - vol.3.  World-Scientific Publishing (Singapore: 2003).  ISBN 981-238-567-3.
     "In this essay I am attempting to give a clear and comprehensive (and comprehensible!) exposition of the formal logic that underlies reductionist treatments of various topics in post-nineteenth-century analytic philosophy.  The aim is to explain in detail--in a number of simple yet instructive cases--how it might happen that talk about some range of putative entities could be meaningful, have truth conditions and so on, even if those entities should be spurious.  Although this ontological position has been adopted in relation to a wide range of putative entities at various times by various people I develop the logical gadgetry here quite specifically in connection with one such move: cardinal and ordinal numbers as virtual objects and always with the Burali-Forti paradox in mind.
     "Such a position (with respect to numbers at least) is one I associate with the work of Quine ('The subtle point is that any progression will serve as a version of number so long and only so long as we stick to one and the same progression.  Arithmetic is, in this sense, all there is to number: there is no saying absolutely what the numbers are; there is only arithmetic.') though I think it is associated in the minds of many others with Dedekind.  Indeed it seems to me to be wider than that, and to be an implicit part of the tradition.   So implicit, and deemed perhaps to be so obvious, that nobody--as far as I know--has bothered to spell it out.  This dereliction has had bad consequences."  From the Preface, p.1.
     "I am no reductionist: for me reductionism is a strategy for flushing out ontological commitment.  I share with the anti-reductionists a hunch that reductionism won't work.  What I do not share is their superstition that it is possible to understand the limitations of reductionist strategies without actually acquiring enough logic to formally execute them.  This is an error: the belief that something won't work is not automatically a reason for not trying it, for even if failures is certain the manner of it might be instructive."  From the Introduction, pp.6-7.
   Content
     Preface
     1. Introduction
     2. Definite Descriptions
     3. Virtual Objects
     4. Cardinal Arithmetic
     5. Iterated Virtuality in Cardinal Arithmetic
     6. Ordinals
     Bibliography
     Index of Definitions
     Index
   
[Fraenkel1922a]
Fraenkel, Abraham A.  Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre.  Mathematische Annalen 86, 230-237.
     This is the article that [Fraenkel1968] cites when discussing Fraenkel's adjustments to Zermelo's system.
   
[Fraenkel1968]
Fraenkel, Abraham A.  Part I.  Historical Introduction.  pp. 1-35 in [Bernays1968].
   
[Gödel1940]
Gödel, Kurt.  The consistency of the axiom of choice and the generalized continuum hypothesis with the axioms of set theory.  Annals of Mathematics Studies, vol. 3.  Lecture notes taken by George W. Brown.  Reprinted with additional notes in 1951 and with further notes in 1966.  Princeton University Press (Princeton, NJ: 1940, 1953, 1966).  ISBN 0-691-07927-7 pbk.  Reprinted in pp. 33-101 of [Gödel1990].
     This is the source for what is known as the [von Neumann-]Bernays-Gödel (BG) set theory.  -- dh:2002-07-25
   
[Gödel1986]
Gödel, Kurt., Feferman, Solomon (editor-in-chief), Dawson, John W. Jr., Kleene, Stephen C., Moore, Gregory H., Solovay, Robert M., van Heijenoort, Jean (eds.).  Kurt Gödel: Collected Works, vol.1: Publications 1929-1936.  Oxford University Press (New York: 1986).  ISBN 0-19-514720-0 pbk.
     "... Each article or closely related group of articles is preceded by an introductory note which elucidates it and places it in historical context.  These notes have been written by the members of the editorial board as well as a number of outside experts [Burton Dreben, A. S. Troelstra, Warren D. Goldfarb, W. V. Quine, Judson Webb, and Rohit Parikh in volume 1]. ...
     "We expect these volumes to be of interest and value to professionals and students in the areas of logic, mathematics, philosophy, history of science, computer science, and even physics, as well as many non-specialist readers with a broad scientific background." -- from the Preface, p.i.
     I have omitted the numerous reviews and have not identified the notes and supplemental contextual material in the full Content.  There are useful notes by Stephen Kleene on achieving consistency of the λ-calculus and Gödel's gradual acceptance of  Church's Thesis in the form of Turing computable functions.   
     One incidental value of this amazing effort is that it provides a standard citation scheme for Gödel's work, and I have incorporated those forms, below.  I have also adopted the citation numbers of [vanHeijenoort1977] for other bibligraphic entries here.
     There is an Addenda and Corrigenda for this volume in the back matter of the second volume.
     I think we are yet to digest the full import of the edifice that Gödel has entrusted to us by his words. -- dh:2002-07-25 
   Content [abridged]
     Preface
     Information for the reader
     Copyright permissions
     Gödel's life and work, Solomon Feferman
     A Gödel chronology, John W. Dawson, Jr.
     On the completeness of the calculus of logic (dissertation) [Gödel1929]
     The completeness of the axioms of the calculus of logic [Gödel1930]
     Some metamathematical results on completeness and consistency [Gödel1930b]
     On formally undecidable propositions of Principia mathematica and related systems I [Gödel1931]
     Discussion on providing a foundation for mathematics [Gödel1931a]
     On the intuitionistic propositional calculus [Gödel1932]
     A special case of the decision problem for theoretical logic [Gödel1932a]
     On completeness and consistency [Gödel1932b]
     A property of the realizations of the propositional calculus [Gödel1932c]
     On independence proofs in the propositional calculus [Gödel1933a]
     On the isometric embeddability of quadruples of points of R3 in the surface of a sphere [Gödel1933b]
     On Wald's axiomatization of the notion of betweenness [Gödel1933c]
     On the axiomatization of the relations of connection in elementary geometry [Gödel1933d]
     On intuitionistic arithmetic and number theory [Gödel1933e]
     An interpretation of the intuitionistic propositional calculus [Gödel1933f]
     Remark concerning projective mappings [Gödel1933g]
     Discussion concerning coordinate-free differential geometry [Gödel1933h]
     On the decision problem for the functional calculus of logic [Gödel1933i]
     On undecidable propositions of formal mathematical systems [Gödel1934]
     On the length of proofs [Gödel1936a]
     Textual Notes
     References
     Index
   
[Gödel1990]
Gödel, Kurt., Feferman, Solomon (editor-in-chief), Dawson, John W. Jr., Kleene, Stephen C., Moore, Gregory H., Solovay, Robert M., van Heijenoort, Jean (eds.).  Kurt Gödel: Collected Works, vol.2: Publications 1938-1974.  Oxford University Press (New York: 1990).  ISBN 0-19-514721-9 pbk.
     This book continues on the model established for the first volume.  Beside the editors, contributors of notes on articles in this volume include S. W. Hawking, Howard Stein, A. S. Troelstra, Judson C. Webb, and Jens Erik Fenstad.  There is an intriguing progression in these later works, as stitched together by the introductory notes.  These stand out for me:
     a. Robert M. Solovay's discussion on the development of consistency results for the axiom of choice (AC) and the generalized continuum hypothesis (GCH) also points out the value of Gödel's formulation of axiomatic set theory, BG (for [von Neumann-]Bernays-Gödel), in [Gödel1940].  
     b. Charles Parsons' introductory note to [Gödel1944] illustrates the degree to which Gödel saw foundational questions for mathematical logic as still unsettled.
     c. Gregory H. Moore's note for [Gödel1947] and its sequel [Gödel1964] provides a sense of the perplexing state of affairs regarding the continuum problem, as we are now left with it.
     d. A. S. Troelstra's note for [Gödel1958] and its [Gödel1972] unpublished revision provide a sense of how investigation is moving into model-theoretic and constructive approaches.  It is plain here and in (a) that explorations of mathematical logic moved into more-abstracted and intricate realms, seemingly far removed from the naive origins of axiomatic set theory.
     e. Solomon Feferman, Robert M. Solovay, and Judson C. Webb riff on the few paragraphs [Gödel1972a] we're given with Gödel's (provisional?) generalizing of the unprovability-of-consistency result, a broadened undecidability theorem, and a perceived error in Turing's reasoning about mental procedures versus mechanical ones.
     -- dh:2002-07-25
   Content
     Preface
     Information for the reader
     Copyright permissions
     List of illustrations [p.188 and 218 entries are reversed]
     The consistency of the axiom of choice and of the generalized continuum hypothesis [Gödel1938]
     The consistency of the generalized continuum hypothesis [Gödel1939]
     Consistency proof for the generalized continuum hypothesis [Gödel1939a]
     The consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set theory [Gödel1940]
     Russell's mathematical logic [Gödel1944]
     Remarks before the Princeton bicentennial conference on problems in mathematics [Gödel1946]
     What is Cantor's continuum problem? [Gödel1947]
     An example of a new type of cosmological solutions of Einstein's field equations of gravitation [Gödel1949]
     A remark about the relationship between relativity theory and idealistic philosophy [Gödel1949a]
     Rotating universes in general relativity theory [Gödel1952]
     On a hitherto unutilized extension of the finitary standpoint [Gödel1958]
     What is Cantor's continuum problem? [Gödel1964]
     On an extension of finitary mathematics which has not yet been used [Gödel1972]
     Some remarks on the undecidability results [Gödel1972a]
     Remark on non-standard analysis [Gödel1974]
     Textual notes
     References
     Addenda and corrigenda to Volume I
     Index
   
[Gödel1995]
Gödel, Kurt., Feferman, Solomon (editor-in-chief)., Dawson, John W. Jr., Goldfarb, Warren., Parsons, Charles., Solovay, Robert M. (eds.).  Kurt Gödel: Collected Works, vol.3: Unpublished essays and lectures.  Oxford University Press (New York: 1995).  ISBN 0-19-514722-7 pbk.
      This book continues on the model established for the first and second volumes.  Beside the editors, contributors of notes on articles in this volume include Cheryl Dawson, Stephen C. Kleene, Israel Halperin, Wilfred Sieg, Martin Davis, A. S. Troelstra, Howard, Stein, David B. Malament, George Boolos, Dagfinn Føllesdal, and Robert Merrihew Adams.  The unpublished works have citations with * in front of the date.  I have abridged the table of contents in the same manner as for the earlier volumes.  -- dh:2002-07-31
     "In this volume, our primary criteria for judging an item to be worthy of inclusion were: (1) the manuscript had to be sufficiently coherent to permit editorial reconstruction; (2) the text was not to duplicate other works substantially in both content and tone (though treatments of similar topics aimed at different audiences or differing in degrees of detail might warrant inclusion); (3) the material had to possess intrinsic scientific interest.
     "Additional justification for our selections came from two lists prepared by Gödel himself, both preserved in his Nachlass and entitled "Was ich publizieren könnte" ("What I could publish"). ...
     "Despite their presence on these lists, we recognize that Gödel did not consider any of the texts presented here to be in final form; before submitting them for publication, he would no doubt have made a number of stylistic, and, in some cases, substantive changes. ...
     "Taken as a whole, the texts presented in this volume substantially enlarge our appreciation of Gödel's scientific and philosophical thought, and in a number of cases they add appreciably to our understanding of his motivations."  -- from the Preface, pp.v-vi.
   Content
     Preface
     Information for the reader
     Copyright permissions
     List of illustrations
     The Nachlass of Kurt Gödel: an overview, John W. Dawson, Jr.
     Gödel's Gabelsberger shorthand, Cheryl A. Dawson
     Lecture on completeness of the functional calculus [Gödel*1930c]
     On undecidable sentences [Gödel*1931?]
     The present situation in the foundations of mathematics [Gödel*1933o]
     Simplified proof of a theorem of Steinitz [Gödel*1933?]
     Lecture at Zilsel's [Gödel*1938a]
     Lecture at Göttingen [Gödel*1939b]
     Undecidable diophantine propositions [Gödel*193?]
     Lecture on the consistency of the continuum hypothesis (Brown University) [Gödel*1940a]
     In what sense is intuitionisstic logic constructive? [Gödel*1941]
     Some observations about the relationship between theory of relativity and Kantian philosophy [Gödel*1946/9]
     Lecture on rotating universes [Gödel*1949b]
     Some basic theorems on the foundations of mathematics and their implications [Gödel*1951]
     Is mathematics syntax of language? [Gödel*1953/9]
     The modern development of the foundation of mathematics in the light of philosophy [Gödel*1961/?]
     Ontological proof [Gödel*1970]
     Some considerations leading to the probable conclusion that the true power of the continuum is aleph-2 [Gödel*1970a]
     A proof of Cantor's continuum hypothesis from a highly plausible axiom about orders of growth [Gödel*1970b]
     Unsent letter to Alfred Tarski [Gödel*1970c]
     Appendix A: Excerpt from *1946/9-A
     Appendix B: Texts related to the ontological proof
     Textual notes
     References
     Addenda and corrigenda to Volumes I and II
     Addendum to this volume
     Index
   
[Goldfarb1995]
Gödel, Kurt., Feferman, Solomon (editor-in-chief)., Dawson, John W. Jr., Goldfarb, Warren., Parsons, Charles., Solovay, Robert M. (eds.).  Kurt Gödel: Collected Works, vol.3: Unpublished essays and lectures.  Oxford University Press (New York: 1995).  ISBN 0-19-514722-7 pbk.  See [Gödel1995]
   
[Halmos1974]
Halmos, Paul Richard.  Naive Set Theory.  Springer-Verlag (New York: 1960, 1974).  ISBN 0-387-90092-6.  Undergraduate texts in mathematics.
     "Every mathematician agrees that every mathematician must know some set theory; the disagreement begins in trying to decide how much is some.  This book contains my answer to that question.  The purpose of the book is to tell the beginning student of advanced mathematics the basic set-theoretic facts of life, and to do so with the minimum of philosophical discourse and logical formalism." -- from the Preface, p. v.
     "The student's task in learning set theory is to steep himself in unfamiliar but essentially shallow generalities till they become so familiar that they can be used with almost no conscious effort.  In other words, general set theory is pretty trivial stuff really, but if you want to be a mathematician, you need some, and here it is; read it, absorb it, and forget it."  -- from the Preface, p. vi.
   Content
     Preface
     1.  The Axiom of Extension
     2.  The Axiom of Specification
     3.  Unordered Pairs
     4.  Unions and Intersections
     5.  Complements and Powers
     6.  Ordered Pairs
     7.  Relations
     8.  Functions
     9.  Families
     10. Inverses and Composites
     11. Numbers
     12. The Peano Axioms
     13. Arithmetic
     14. Order
     15. The Axiom of Choice
     16. Zorn's Lemma
     17. Well Ordering
     18. Transfinite Recursion
     19. Ordinal Numbers
     20. Sets of Ordinal Numbers
     21. Ordinal Arithmetic
     22. The Schröder-Bernstein Theorem
     23. Countable Sets
     24. Cardinal Arithmetic
     25. Cardinal Numbers
     Index
   
[Holmes1998]
Holmes, M.Randall.  Elementary Set Theory with a Universal Set.  Université catholique de Louvain Département de Philosophie, Cahiers du Centre de Logique v.10.  Academia Bruylant (Louvain-la-Neuve, Belgium: 1998).  ISBN 2-87209-488-1 pbk.
   Content
     1. Introduction: Why Save the Universe?
     2. The Set Concept
     3. Boolean Operations on Sets
     4. Building Finite Structures
     5. The Theory of Relations
     6. Sentences and Sets
     7. Stratified Comprehension
     8. Philosophical Interlude
     9. Equivalence and Order
     10. Introducing Functions
     11. Operations on Functions
     12. The Natural Numbers
     13. The Real Numbers
     14. The Axiom of Choice
     15. Ordinal Numbers
     16. Cardinal Numbers
     17. Three Theorems
     18. Sets of Real Numbers
     19. Strongly Cantorian Sets
     20. Well-Founded Extensional Relations
     21. Surreal Numbers
     22. The Structure of the Transfinite
     23. Stratified Lambda-Calculus
     24. Acknowledgements and Notes
     Appendix: Selected Axioms and Theorems
     Bibliography
     Index
   
[Jeffrey1998]
Jeffrey, Richard.  editor for [Boolos1998].
   
[Jeffrey2002]
Boolos, George S., Burgess, John P., Jeffrey, Richard.  Computability and Logic. ed.4.  Cambridge University Press (Cambridge: 2002).  ISBN 0-521-00758-5 pbk.  See [Boolos2002]
   
[Jourdain1915]
Cantor, Georg.  Contributions to the Founding of the Theory of Transfinite Numbers.  Translation, Introduction and Notes by Philip E.  B. Jourdain.  Open Court (London: 1915). Unabridged republished edition by Dover Publications (New York: 1955).  ISBN 0-486-60045-9 pbk.  See [Cantor1915]
   
[Kleene1986]
Gödel, Kurt., Feferman, Solomon (editor-in-chief), Dawson, John W. Jr., Kleene, Stephen C., Moore, Gregory H., Solovay, Robert M., van Heijenoort, Jean (eds.).  Kurt Gödel: Collected Works, vol.1: Publications 1929-1936.  Oxford University Press (New York: 1986).  ISBN 0-19-514720-0 pbk.  See [Gödel1986]
   
[Kleene1990]
Gödel, Kurt., Feferman, Solomon (editor-in-chief), Dawson, John W. Jr., Kleene, Stephen C., Moore, Gregory H., Solovay, Robert M., van Heijenoort, Jean (eds.).  Kurt Gödel: Collected Works, vol.2: Publications 1938-1974.  Oxford University Press (New York: 1990).  ISBN 0-19-514721-9 pbk.  See [Gödel1990]
   
[Mendelson1970]
Stolyar, Abram Aronovich.  Introduction to Elementary Mathematical Logic.  Dover (New York: 1970).  ISBN 0-486-64561-4 pbk.  Unabridged and unaltered 1983 republication of the work published by MIT Press (Cambridge, MA: 1970).  Translation of Elementarnoe vvedenie v matematicheskuiu logiku, Prosveshcheniye Press (Moscow: 1965), with translation from the Russian edited by Elliot Mendelson.  See [Stolyar1970]
   
[Mendelson1997]
Mendelson, Elliott.  Introduction to Mathematical Logic.  ed.4.  Chapman & Hall/CRC (Boca Raton, FL: 1964, 1979, 1987, 1997).  ISBN 0-412-80830-7.
     "This is a compact introduction to some of the principal topics of mathematical logic.  In the belief that beginners should be exposed to the easiest and most natural proofs, I have used free-swinging set-theoretic methods.  The significance of a demand for constructive proofs can be evaluated only after a certain amount of experience with mathematical logic has been obtained.  If we are to be expelled from ‘Cantor's paradise’ (as non-constructive set theory was called by Hilbert), at least we should know what we are missing. ...
     "I believe that the essential parts of the book can be read with ease by anyone with some experience in abstract mathematical thinking.  There is, however, no specific prerequisite."  -- from the Preface, p.ix.
     "Although logic is basic to all studies, its fundamental and apparently self-evident character discouraged any deep logical investigations until the late 19th century.  Then, under the impetus of the discovery of non-Euclidean geometry and the desire to provide a rigorous foundation for calculus and higher analysis, interest in logic revived.  This new interest, however, was still rather unenthusiastic until, around the turn of the century, the mathematical world was shocked by the discovery of the paradoxes--that is, arguments that lead to contradictions.  The most important paradoxes are described here. ...
     "Whatever approach one takes to the paradoxes, it is necessary first to examine the language of logic and mathematics to see what symbols may be used, to determine the ways in which these symbols are put together to form terms, formulas, sentences and proofs, and to find out what can and cannot be proved if certain axioms and rules of inference are assumed.  This is one of the tasks of mathematical logic, and, until it is done, there is no basis for comparing rival foundations of logic and mathematics.  The deep and devastating results of Gödel, Tarski, Church, Rosser, Kleene, and many others have been ample reward for the labour invested and have earned for mathematical logic its status as an independent branch of mathematics."  -- from the Introduction, pp.1, 4-5.
     This is a weighty book. In many ways it provides more depth on the standard topics, and connections to a variety of inter-related topics that are valuable under the covers of one book, arranged with consistent attention.  This is also a high-density book.  I was amazed to see how little white space there is in the 10-page introduction.  Don't worry, the pages have margins and there is more use of layout in the main chapters.  And there is a certain economy and intensity that leads me to have alternative coverage of common-ground topics, like Enderton's book, to fall back on when I hit a pot hole in my excursions here.  -- dh:2002-07-17
     I give the expanded contents of the last three chapters for the benefit of computer scientists who want to comprehend the bearing that mathematical logic has on their field.
   Content
     Preface
     Introduction
     1. The Propositional Calculus
     2. Quantification Theory
     3. Formal Number Theory
          3.1 An axiom system
          3.2 Number-theoretic functions and relations
          3.3 Primitive recursive and recursive functions
          3.4 Arithmetization.  Gödel numbers
          3.5 The fixed-point theorem.  Gödel's incompleteness theorem
          3.6 Recursive undecidability.   Church's theorem
     4. Axiomatic Set Theory
          4.1 An axiom system
          4.2 Ordinal numbers
          4.3 Equinumerosity.  Finite and denumerable sets
          4.4 Hartogs' Theorem.   Initial ordinals.  Ordinary arithmetic
          4.5 The axiom of choice.  The axiom of regularity
          4.6 Other axiomatizations of set theory
     5. Computability
          5.1 Algorithms.  Turing machnes
          5.2 Diagrams
          5.3 Partial recursive functions.  Unsolvable problems
          5.4 The Kleene-Mostovski hierarchy.  Recursively enumerable sets
          5.5 Other notions of computability
          5.6 Decision problems
     Appendix.  Second Order Logic
     Answers to Selected Exercises
     Bibliography
     Notation
     Index
   
[Moore1986]
Gödel, Kurt., Feferman, Solomon (editor-in-chief), Dawson, John W. Jr., Kleene, Stephen C., Moore, Gregory H., Solovay, Robert M., van Heijenoort, Jean (eds.).  Kurt Gödel: Collected Works, vol.1: Publications 1929-1936.  Oxford University Press (New York: 1986).  ISBN 0-19-514720-0 pbk.  See [Gödel1986]
   
[Moore1990]
Gödel, Kurt., Feferman, Solomon (editor-in-chief), Dawson, John W. Jr., Kleene, Stephen C., Moore, Gregory H., Solovay, Robert M., van Heijenoort, Jean (eds.).  Kurt Gödel: Collected Works, vol.2: Publications 1938-1974.  Oxford University Press (New York: 1990).  ISBN 0-19-514721-9 pbk.  See [Gödel1990]
   
[Mostowski1953]
Mostowski, Andrzej., Robinson, Raphael M., Tarski, Alfred.  II. Undecidability and Essential Undecidability in Arithmetic.  Part II of [Tarski1953].
   
[Nelson1986]
Nelson, Edward.  Predicative Arithmetic.  Mathematical Notes 32.  Princeton University Press (Princeton, NJ: 1986).  ISBN 0-691-08455-6 pbk.
     "The reason for mistrusting the induction principle is that it involves an impredicative concept of number.  It is not correct to argue that induction only involves the number from 0 to n [when establishing that θ(n) implies θ(n+1)]; the property of n being established may be a formula with bound variables that are thought of as ranging over all numbers.  That is, the induction principle assumes that the natural number system is given.  ...
     "It appears to be universally taken for granted by mathematicians, whatever their views on foundational questions may be, that the impredicativity inherent in the induction principle is harmless--that there is a concept of number given in advance of all mathematical constructions, that discourse within the domain of numbers is meaningful.  But numbers are symbolic construction; a construction does not exist until it is made; when something new is made, it is something new and not a selection from a pre-existing collection.  There is no map of the world because the world is coming into being.
     "Let us explore the possibility of developing arithmetic predicatively."  -- from section 1, The impredicativity of induction, pp. 1-2.
   Content
     Acknowledgments
     1. The impredicativity of induction
     2. Logical terminology
     3. The axioms of arithmetic
     4. Order
     5. Induction by relativization
     6. Interpretability in Robinson's theory
     7. Bounded induction
     8. The bounded least number principle
     9. The Euclidean algorithm
     10. Encoding
     11. Bounded separation and minimum
     12. Sets and functions
     13. Exponential functions
     14. Exponentiation
     15. A stronger relativization scheme
     16. Bounds on exponential functions
     17. Bounded replacement
     18. An impassable barrier
     19. Sequences
     20. Cardinality
     21. Existence of sets
     22. Semibounded replacement
     23. Formulas
     24. Proofs
     25. Derived rules of inference
     26. Special  constants
     27. Extensions by definition
     28. Interpretations
     29. The arithmetization of arithmetic
     30. The consistency theorem
     31. Is exponentation total?
     32. A modified Hilbert program
     Bibliography
     General index
     Index of defining axioms
   
[Parsons1995]
Gödel, Kurt., Feferman, Solomon (editor-in-chief)., Dawson, John W. Jr., Goldfarb, Warren., Parsons, Charles., Solovay, Robert M. (eds.).  Kurt Gödel: Collected Works, vol.3: Unpublished essays and lectures.  Oxford University Press (New York: 1995).  ISBN 0-19-514722-7 pbk.  See [Gödel1995]
   
[Quine1969]
Quine, Willard Van Orman.  Set Theory and Its Logic.  Revised edition.  Harvard University Press (Cambridge, MA: 1963, 1969).  ISBN 0-674-80207-1 pbk.
     "Set theory is the mathematics of classes.  Sets are classes.  The notion of class is so fundamental to thought that we cannot hope to define it in more fundamental terms.  We can say that a class is an aggregate, any collection, any combination of objects of any sort; if this helps, well and good.  But even this will be less help than hindrance unless we keep clearly in mind that the aggregating or collecting or combining here is to connote no actual displacement of the objects, any further than the aggregation or collection or combination of say seven given pairs of shoes is not to be identified with the aggregation or collection or combination of those fourteen shoes, nor with that of the twenty-eight soles and uppers.  In short, a class may be thought of as an aggregate or collection or combination of objects just so long as 'aggregate' or 'collection' or 'combination' is understood strictly in the sense of 'class'." From the Introduction, p.1.
     After taking great pains to compare various approaches, including those of Zermelo-Fraenkel and Bernays-von Neumann, and the contingent results on their consistency, Quine ends with this parting shot: "Let me seize this opportunity to leave the reader with a sense of how open the problem of the best foundation for set theory remains." p.329.
   Content
     Preface to the Revised Edition
     Preface to the First Edition
     Introduction
    Part One.  The Elements
     I. Logic
     II. Real Classes
     III. Classes of Classes
     IV. Natural Numbers
     V. Iteration and Arithmetic
    Part Two.  Higher Forms of Number
     VI. Real Numbers
     VII. Order and Ordinals
     VIII. Transfinite Recursion
     IX. Cardinal Numbers
     X. The Axiom of Choice
    Part Three.  Axiom Systems
     XI. Russell's Theory of Types
     XII. General Variables and Zermelo
     XIII. Stratification and Ultimate Classes
     XIV. Von Neumann's System and Others
     Synopsis of Five Axiom Systems
     List of Numbered Formulas
     Bibliographical References
     Index
   
[Quine1977]
Quine, W. V.  Introduction to [Schönfinkel1924] "On the Building Blocks of Mathematical Logic," pp.355-357 in From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, 3rd. printing [vanHeijenoort1977].
   "The initial aim of the paper is reduction of the number of primitive notions of logic.  ... Then presently, the purpose deepens.  It comes to be nothing less than the general elimination of variables."
   There is, in particular, a strong desire to escape the use of quantifiers and instead determne class membership with predicates (i.e., functions), avoiding or at least significantly-reducing presumption of (transfinite) entireties.
   "Schönfinkel's notions, which do suffice to do the work of quantifiers and their variables altogether, go far beyond ... .  The crux of the matter is that Schönfinkel lets functions stand as arguments."
   This remarkable provision seems to escape the type theories of [Whitehead & Russell] and will lead to a different notion, emerging as functional types, with Category Theory perhaps taking over the type-hierarchy burden.  That this improvement is interpretable in realizable models of computation on denumerable entities is remarkable in turn.  dh:2024-07-29
 
[Quine1980]
Quine, Willard Van Orman.  Elementary Logic.  Revised edition.  Harvard University Press (Cambridge, MA: 1941, 1965, 1980).  ISBN 0-674-24451-6 pbk.
     This may be more elementary than I had been seeking.  Having been chided to dig into FOL and demonstrate some mastery of it, I went looking for something that would take me gently up through the complete results.  This is not that book.  I think it might take me to the threshold in a gentle way, however.  Albeit gentler, I am of the mind that Methods of Logic, provides superior coverage for those wanting to understand the overall reach of predicate logics.  I have expanded the content of the last part of Elementary Logic to assist in that determination. [dh:2004-02-13]
     "This little book provides a single strand of simple techniques for the central business of modern logic, seldom looking to the right or to the left for alternative methods or peripheral problems.  Basic formal concepts are explained, the paraphrasing of words into symbols is treated at some length, a testing procedure is given for truth-function logic, and a complete proof procedure is given for the logic of quantifiers.  At the end there are brief glimpses of further matters."  Preface to the Revised Edition, p.vii.
     "To say that mathematics in general has been reduced to logic hints of some new firming up of mathematics at its foundations.  This is misleading.  Set theory is less settled and more conjectural than the classical mathematical superstructure that can be founded upon it.  These infirmities of set theory are themselves good reason to see set theory as an extralogical department of mathematics.  Logic in the best and narrowest sense has all the firmness and dependability that its name connotes.  Reality being what it is, we cannot expect most truths to admit of foundation purely within logic in such a sense."  §48 Membership, p.125.
     I shall continue to compile citations on set theory under the topic of logic in honor of custom and laziness.  [dh:2004-02-12]
   Content
     Preface, 1980
     Preface to the Revised Edition [1965]
     Preface to the 1941 Edition
          1. Introduction
     I. Statement Composition
     II. Truth-Functional Transformations
     III. Quantification
     IV. Quantificational Inference
          39. Quantificational Schemata
          40. Predicates
          41. Restraints on Introducing
          42. Substitution Extended
          43. Validity Extended
          44. Equivalence Extended
          45. Inconsistency Proofs
          46. Logical Arguments
          47. Identity and Singular Terms
          48. Membership
     Index
   
[Quine1980b]
Quine, Willard Van Orman.  From a Logical Point of View: Nine Logico-Philosophical Essays.  Second Edition, revised. Harvard University Press (Cambridge, MA: 1953, 1961, 1980).  ISBN 0-674-32351-3 pbk.
     I have this book for at least two reasons.  It has "On what there is" and, in addition, the formulation of New Foundations that is often discussed and not so well described.  In this material, Quine has not yet surrendered the Principia Mathematica notations and concepts that are reluctantly surrendered in the fourth edition of Methods of Logic.  For a comparison, Quine's Elementary Logic, Set Theory and Its Logic, and and Mathematical Logic preserve their Whitehead and Russell influence. 
     For Quine's own appraisal of the relative merits of ZF, NF, ML, and von Neumann's approach, it is useful to consult the supplementary material in Chapter V and the 1980 Foreword.
     With regard to the notion of inherent meaning, Quine points out that some of the essays "reflected a dim view of the notion of meaning.  A discouraging response from somewhat the fringes of philosophy has been that my problem comes of taking words as bare strings of phonemes rather than seeing that they are strings with meaning.  Naturally, they say, if I insist on meaningless strings I shall be at  a loss for meanings.  They fail to see that a bare and identical string of phonemes can have a meaning, or several, in one or several languages, through its use by sundry people or peoples, much as I can have accounts in several banks and relatives in several countries without somehow containing them or being several persons. ... I hope this paragraph has been superfluous for most readers."  Foreword, 1980, p.viii.
     With regard to the connection of logic with language and meaning, the last essay has much to offer: "We freed ourselves, six paragraphs back, of any general constraint to admit the inference of ‘a exists’ from ‘Fa’ and ‘~Fa’.  We are led to wonder, however, just what statements containing ‘a’ should be required as requiring for their truth that a exist."  Meaning and existential inference, p.164.
   Content
     Foreword, 1980
     Preface to the Second Edition [1962]
     Preface
     I. On what there is [1948]
     II. Two dogmas of empiricism [1951]
     III. The problem of meaning in linguistics [1951]
     IV. Identity, ostension, and hypostasis [1950]
     V. New foundations for mathematical logic [1937]
     VI. Logic and the reification of universals [1939, 1947, 1950]
     VII. Notes on the theory of reference [1951]
     VIII. Reference and modality [1943, 1947, 1952]
     IX. Meaning and existential reference [1947, 1953]
     Origins of the Essays
     Bibliographical references
     Index
   
[Quine1981]
Quine, Willard Van Orman.  Mathematical Logic.  revised edition.  Harvard University Press (Cambridge, MA: 1940, 1951, 1979, 1981).  ISBN 0-674-55451-5 pbk.
     It would appear that there is little change between this and the 1951 edition.  The original work endures.  
     This work is solidly grounded in the Principia Mathematica model, with great improvement in clarity:
     "I like to think that I also helped in the struggle against confusions of use and mention.  Certainly I was persistent in precept and scrupulous in example, and happily the situation has vastly improved.  Still, my efforts against the related confusion of implication with the conditional, and of equivalence with what I named the biconditional, have been only partially successful if we may judge by lingering nomenclature.
     "The final chapter has two purposes.  One purpose was to show how the metalogical discourse about a formal system would look when formalized in turn.  The other and higher purpose was to teach a proof of Gödel's incompleteness theorem.  My proof, unlike Gödel's shuns numbers until the coup de grâce.  It proceeds rather in what I called protosyntax, as a proof that protosyntactical truth is not definable in protosyntax.  Thus seen, it is a case of Tarski's theorem that theories meeting certain reasonable conditions cannot contain their own truth definitions.  Finally, Gödel's theorem is derived by correlating protosyntax with number theory."  -- From the Preface, 1981, p.v.
     To provide a sense of the foundational work around formalization and the use of language and metalanguage, I display here the complete content for Chapters One - Three and for Chapter Seven.  This is an useful contrasting to the mathematical logics of Church and Curry, and another backdrop to the modern conceptualization in Mendelson. -- dh:2002-07-25.
   Content
     Preface, 1981
     Preface to the Revised Edition (1951)
     Preface
     Introduction
     Chapter One.  Statements
          1. Conjunction, Alternation, and Denial
          2. The Conditional
          3. Iterated Composition
          4. Use Versus Mention
          5. Statements about Statements
          6. Quasi-Quotation
          7. Parentheses and Dots
          8. Reduction to Three Primitives
          9. Reduction to One Primitive
          10. Tautology
          11. Selected Tautologous Forms
     Chapter Two.  Quantification
          12. The Quantifier
          13. Formulae
          14. Bondage, Freedom, Closure
          15. Axioms of Quantification
          16. Theorems
          17. Metatheorems
          18. Substitutivity of the Biconditional
          19. Existential Quantification
          20. Distribution of Quantifiers
          21. Alphabetic Variance
     Chapter Three.  Terms
          22. Class and Member
          23. Logical Formulae
          24. Abstraction
          25. Identity
          26. Abstraction Resumed
          27. Descriptions and Names
     Chapter Four.  Extended Theory of Classes
     Chapter Five.  Relations
     Chapter Six.  Number
     Chapter Seven.  Syntax
          53. Formality
          54. The Syntactical Primitive
          55. Protosyntax
          56. Formula and Matrix Defined
          57. Axioms of Quantification Defined
          58. Theorem Defined
          59. Protosyntax Self-Applied
          60. Incompleteness
     Appendix.  Theorem versus Metatheorem
     List of Definitions
     List of Theorems and Metatheorems
     Bibliographical References
     Index of Proper Names
     Index of Subjects
   
[Quine1982]
Quine, Willard Van Orman.  Methods of Logic.  Fourth Edition.  Harvard University Press (Cambridge, MA: 1959, 1972, 1978, 1982).  ISBN 0-674-57176-2 pbk.
     I am looking for a book that works through predicate logic to the point where its application can be appreciated and the relationships among First-Order Logic (FOL), Higher-Order Logics (HOLs), Peano Arithmetic (PA) toward number theory, and set theory can be appreciated and differentiated where clarity is important.  This seems like a friendly place to commence such a journey.
     I must confess that another incentive for me was to find Quine's discussion of the Geach-Kaplan sentence on p.293.  For me, this just compounds my bafflement about how "Some people admire only one another" is translated into an HOL reading. [dh:2004-02-13]
     "Logic, like any science has as its business the pursuit of truth.  What are true are certain statements; and the pursuit of truth is the endeavor to sort out the true statements from the others, which are false. ... But scientific activity is not the indiscriminate amassing of truths; science is selective and seeks the truths that count for most, either in point of intrinsic interest or as instruments for coping with the world."  Introduction, p.1.
   Content
     Preface [1982]
     Introduction
     Part I. Truth Functions
     Part II. General Terms and Quantifiers
     Part III. General Theory of Quantification
          27. Schemata Extended
          28. Substitution Extended
          29. Pure Existentials
          30. The Main Method
          31. Application
          32. Completeness
          33. Löwenheim's Theorem
          34. Decisions and the Undecidable
          35. Functional Normal Forms
          36. Herbrand's Method
          37. Other Methods of Validity
          38. Deduction
          39. Soundness
          40. Deductive Strategy
     Part IV. Glimpses Beyond
          41. Singular Terms
          42. Identity
          43. Descriptions
          44. Elimination of Singular Terms
          45. Elimination of Variables
          46. Classes
          47. Number
          48. Axiomatic Set Theory
     Partial Answers to Exercises
     Bibliography
     Index
   
[Quine1986]
Quine, Willard Van Orman.  Philosophy of Logic.  ed.2.  Harvard University Press (Cambridge, MA: 1970, 1986).  ISBN 0-674-66563-5.
     "If pressed to [provide] a discursive definition of [deductive logic], I would say that logic is the systematic study of the logical truths.  Pressed further, I would say to read this book.
     "Since I see logic as the resultant of two components, truth and grammar, I shall treat truth and grammar prominently.  But I shall argue against the doctrine that the logical truths are true because of grammar, or because of language.
     "The notions of proposition and meaning will receive adverse treatment.  Set theory will be compared and contrasted with logic, and ways will be examined of disguising each to resemble the other.  The status and claims of alternative logic will be discussed, and reasons will be adduced for being thankful for what we have."  -- from the Preface, 1986, p.vii.
   Content
     Preface, 1986
     1. Meaning and Truth
     2. Grammar
     3. Truth
     4. Logical Truth
     5. The Scope of Logic
     6. Deviant Logics
     7. The Ground of Logical Truth
     For Further Reading
     Index
   
[Robinson1953]
Mostowski, Andrzej., Robinson, Raphael M., Tarski, Alfred.  II. Undecidability and Essential Undecidability in Arithmetic.  Part II of [Tarksi1953].
   
[Robinson1996]
Robinson, Abraham.  Non-Standard Analysis.  ed.2.  Princeton University Press (Princeton, NJ: 1965, 1973, 1996).  ISBN 0-691-04490-2 pbk.  Re-issue of the 1973 second edition with a 1996 foreword by Wilhelmus A. J. Luxemburg.
     This book is about mathematics, in particular analysis, based on a non-standard model of numbers in which infinitesimal and infinitely large elements have first-class standing as individuals of the theory.  A highly-productive approach grounded in mathematical logic, this book provides an application of model-theoretic approaches.  The chapter on logic supports other materials on mathematical logic.  More information is under Readings in Mathematics.
   
[Rogers1987]
Rogers, Hartley, Jr.  Theory of Recursive Functions and Effective Computability.  MIT Press (Cambridge, MA: 1967, 1987).  ISBN 0-262-68052-1 pbk.
   Content
     Preface to Paperback Edition (1987)
     Introduction: Prerequisites and Notation
     Emendations
     1. Recursive Functions
     2. Unsolvable Problems
     3. Purposes; Summary
     4. Recursive Invariance
     5. Recursive and Recursively Enumerable Sets
     6. Reducibilities
     7. One-One Reducibility; Many-One Reducibility; Creative Sets
     8. Truth-Table Reducibilities; Simple Sets
     9. Turing Reducibility; Hypersimple Sets
     10. Posts's Problem
     11. The Recursion Theorem
     12. Recursively Enumerable Sets as a Lattice
     13. Degrees of Unsolvability
     14. The Arithmetical Hierarchy (Part 1)
     15. The Arithmetical Hierarchy (Part 2)
     16. The Analytical Hierarchy
     Bibliography
     Index of Notations
     Subject Index
   
[Rosenbloom1950]
Rosenbloom, Paul.  The Elements of Mathematical Logic.  Dover (New York: 1950).  pbk.
     I wore out my original copy of this book, acquired sometime before 1962.  Although now out-of-print, a replacement lives up to Dover's promise that this is a permanent book. [dh: 2000-10-11]
     "In this book we shall study the laws of logic by mathematical methods.  This may seem unfair, since logic is used in constructing mathematical proofs, and it might appear that the study of logic should come before the study of mathematics.  Such a procedure is, however, typical of science.  Our actual knowledge is a narrow band of light flanked on both sides by darkness.  We may, on the one hand, go forward and develop further the consequences of known principles.  Or else we may press backward the obscurity in which the foundations of science are enveloped.  Just by using mathematical methods ... we can throw new and important light on the logical principles used in mathematics.  This approach has led to more knowledge about logic in one century than had been obtained from the death of Aristotle up to 1847, when Boole's masterpiece was published."  From the Introduction, p.i.
   Contents
     Introduction
     I. The Logic of Classes
          1. Informal introduction.  Fundamental theorems
          2. Boolean algebra as a deductive science
          3. The structure and representation of Boolean algebras
     II. The Logic of Propositions
          1. Fundamentals
          2. Alternative formulations
          3. Deductive systems
          4. Many valued logics, modal logics, intuitionism
     III. The Logic of Propositional Functions
          1. Informal introduction
          2. The functional logic of the first order
          3. Some very expressive languages
          4. Combinatory logics
          5. The development of mathematics within an object language
          6. The paradoxes
          7. The axiom of choice
     IV. The General Syntax of Language
          1. Basic concepts.  Simple languages
          2. Production, canonical languages, extension, and definition
          3. Normal languages.  Theorems of Post and Gödel
     Appendix 1.  Canonical forms of L1, L2', and Lz
     Appendix 2.  Algebraic approach to languages.  Church's theorem.
     Bibliographical and Other Remarks
     Index
   
[Russell1937]
Russell, Bertrand.  The Principles of Mathematics.  ed. 2. George Allen & Untwin Ltd. (London: 1903, 1937).
     I have heard it asked: "What was Cantor trying to do when he came up with transfinite numbers (the cardinals for the powers of infinite sets)?"  What he was trying to deal with was problems of reasoning about the infinite that came up in everything beyond the natural numbers: reals, spaces of reals, measures, quantities, the foundations of mathematical physics.  
     What was at stake can be seen in Russell's ambitious table of contents.  It is not that Russell saw these as resolved by mathematical logic, though he would have though such to be achievable at least until Gödel unseated the undertaking.  I don't take that to mean that the effort was not worthwhile, although the laborious formulation of Principia Mathematica has been found to be avoidable within the more succinct approaches of present-day logic with axiomatic set theory.  
     We have not ceased to rely on the logic and mathematics of the unattainable, nor did Gödel compel us to.  The situation at hand is far more subtle than the black-and-white resolution we yearn for. Russell's 1937 introduction lays out the dilemma for all to see, no matter which direction we might favor and what of Russell's particular spin we choose to neglect.  The inquiry continues into its second century.  This provides useful background to subsequent commentators.  It is interesting that, however much the details are refined and the reasoning improved and streamlined, the great questions remain.  -- dh:2002-07-27
     "The fundamental thesis of the following pages, that mathematics and logic are identical, is one which I have never since seen any reason to modify.  This thesis was, at first, unpopular ... but such feelings would have no lasting influence if they had been unable to find support in more serious reasons for doubt.  These reasons are, broadly speaking, of two opposite kinds: first, that there are certain unsolved difficulties in mathematical logic, which make it appear less certain than mathematics is believed to be; and secondly that, if the logical basis of mathematics is accepted, it justifies, or tends to justify, much work, such as that of Georg Cantor, which is viewed with suspicion by many mathematicians on account of the unsolved paradoxes which it shares with logic.  These two opposite lines of criticism are represented by the formalists, led by Hilbert, and the intuitionists, led by Brouwer.
     "The formalist interpretation ... consists in leaving the integers undefined, but asserting concerning them such axioms as shall make possible the deduction of the usual arithmetical propositions.  That is to say, we do not assign any meaning to our symbols 0, 1, 2, . . except that they are to have certain properties enumerated in the axioms.  ... The later integers may be defined when 0 is given, but 0 is to be merely something having the assigned characteristics.  Accordingly the symbols 0, 1, 2, ... do not represent one definite series, but any progression whatever.   The formalists have forgotten that numbers are needed. ... For the symbol '0' may be taken to mean any finite integer, without thereby making any of Hilbert's axioms false; and thus every number-symbol becomes infinitely ambiguous.  The formalists are like a watchmaker who is so absorbed in making his watches look pretty that he has forgotten their purpose of telling the time, and has therefore omitted to insert any works.
     "There is another difficulty ... and that is as regards existence.  ... For [Hilbert] 'existence' as usually understood, is an unnecessary metaphysical concept, which should be replaced by the precise concept of non-contradiction. ... Here again, he has forgotten that arithmetic has practical uses.  ... Our reasons for being specially interested in the axioms that lead to ordinary arithmetic lie outside arithmetic, and have to do with the application of numbers to empirical material.  This application itself forms no part of either logic or arithmetic; but a theory which makes it a priori impossible cannot be right. ... 
     "The intuitionist theory ... is a more serious matter. ... The essential point here is the refusal to regard a proposition as either true or false unless some method exists of deciding the alternative.  Brouwer denies the law of excluded middle where no such method exists.  This destroys, for example, the proof that there are more real numbers than rational numbers, and that, in the series of real numbers, every progression has a limit.  Consequently large parts of analysis, which for centuries have been thought well established, are rendered doubtful." -- from the Introduction to the Second Edition, pp.v-vi.
     2004-12-02: "Pure mathematics is the class of all propositions of the form 'p implies q,' where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants.  And logical constants are all notions defined in terms of the following: Implication, the relation of a term to a class of which it is a member, the notion of such that, the notion of relation, and such further notions as may be involved in the general notion of propositions of the above form.  In addition to these, mathematics uses a notion which is not a constituent of the propositions which it considers, namely the notion of truth." -- Chapter I, p.3.
   Content
     Introduction to the Second Edition
     Preface
     Part I.  The Indefinables of Mathematics
          Chapter I.  Definition of Pure Mathematics
          Chapter II.  Symbolic Logic
          Chapter III.  Implication and Formal Implication
          Chapter IV.  Proper Names, Adjectives and Verbs.
          Chapter V.  Denoting
          Chapter VI.  Classes
          Chapter VII.  Propositional Functions
          Chapter VIII.  The Variable
          Chapter IX.  Relations
          Chapter X.  The Contradiction
     Part II.  Number
          Chapter XI.  Definition of Cardinal Numbers
          Chapter XII.  Addition and Multiplication
          Chapter XIII.  Finite and Infinite
          Chapter XIV.  Theory of Finite Numbers
          Chapter XV.  Addition of Terms and Addition of Classes
          Chapter XVI.  Whole and Part
          Chapter XVII.  Infinite Wholes
          Chapter XVIII.  Ratios and Fractions
     Part III.  Quantity
          Chapter XIX.  The Meaning of Magnitude
          Chapter XX.  The Range of Quantity
          Chapter XXI.  Numbers as Expressing Magnitudes: Measurement
          Chapter XXII.  Zero
          Chapter XXIII.  Infinity, the Infinitesimal, and Continuity
     Part IV.  Order
          Chapter XXIV.  The Genesis of Series
          Chapter XXV.  The Meaning of Order
          Chapter XXVI.  Asymmetrical Relations
          Chapter XXVII.  Difference of Sense and Difference of Sign
          Chapter XXVIII.  On the Difference Between Open and Closed Series
          Chapter XXIX.  Progressions and Ordinal Numbers
          Chapter XXX.  Dedekind's Theory of Number
          Chapter XXXI.  Distance
     Part V.  Infinity and Continuity
          Chapter XXXII.  The Correlation of Series
          Chapter XXXIII.  Real Numbers
          Chapter XXXIV.  Limits and Irrantional Numbers
          Chapter XXXV.  Cantor's First Definition of Continuity
          Chapter XXXVI.  Ordinal Continuity
          Chapter XXXVII.  Transfinite Cardinals
          Chapter XXXVIII.  Transfinite Ordinals
          Chapter XXXIX.  The Infinitesimal Calculus
          Chapter XL.  The Infinitesimal and the Improper Infinite
          Chapter XLI.  Philosophical Arguments Concerning the Infinitesimal
          Chapter XLII.  The Philosophy of the Continuum
          Chapter XLIII.  The Philosophy of the Infinite
    Part VI.  Space
          Chapter XLIV.  Dimensions and Complex Numbers
          Chapter XLV.  Projective Geometry
          Chapter XLVI.  Descriptive Geometry
          Chapter XLVII.  Metrical Geometry
          Chapter XLVIII.  Relation of Metrical to Projective and Descriptive Geometry
          Chapter XLVIX.  Definitions of Various Spaces
          Chapter L.  The Continuity of Space
          Chapter LI.  Logical Arguments against Points
          Chapter LII.  Kant's Theory of Space
     Part VII.  Matter and Motion
          Chapter LIII.  Matter
          Chapter LIV.  Motion
          Chapter LV.  Causality
          Chapter LVI.  Definition of a Dynamical World
          Chapter LVIL.  Newton's Laws of Motion
          Chapter LVIII.  Absolute and Relative Motion
          Chapter LIX.  Hertz's Dynamics
     Appendix A.  The Logical and Arithmetical Doctrines of Frege
     Appendix B.  The Doctrine of Types
     Index
   
[Russell1997b]
Whitehead, Alfred North., Russell, Bertrand.  Principia Mathematica to *56.  Cambridge Mathematical Library edition.  Cambridge University Press (London: 1910, 1927, 1962, 1997).  ISBN 0-521-62606-4 pbk.  See [Whitehead1997].
   
[Schönfinkel1924]
Schönfinkel, Moses.  Über die Bausteine der mathematischen Logik.  Mathematische Annalen 92 (1924), 305-316.  Stefan Bauer-Mengelberg English translation "On the Building Blocks of Mathematical Logic" reprinted in [vanHeijenoort1977: 355-366] with an introduction by W.V.Quine [Quine1977].
  
[Smullyan1961]
Smullyan, Raymond M.  Theory of Formal Systems.   Annals of Mathematical Studies Number 47. Princeton University Press (Princeton, NJ: 1961).  ISBN 0-691-08047-X pbk.
     It is difficult to believe that I picked this small book off of a shelf at Seattle's University Bookstore when it first appeared, in its now-quaint typewritten manuscript.  I did not expect to find it still in print, until I had occasion to check after recommending its treatment of formal systems from somewhere near the ground up.  I am delighted to have it again in my possession.  There is much here that provides value today.  dh:2004-06-15
     "We are interested in first giving a precise definition of a 'formal' or 'finitary' mathematical system, and then in establishing an important theorem, based on the works of Church and Post, concerning an interesting limitation of our possible knowledge of such systems. ...
     "Our plan is to define first a certain type of system termed an 'elementary formal system' which will serve as a basis for our study of metamethematics [sic] (theory of mathematical systems).  The general notion of a 'formal system' will then be defined in terms of these elementary systems." - Chapter I, p.1.
   Content
     Annals of Mathematics Studies [bonus list of the first 47]
     Preface
     I. Formal Mathematical Systems
     II. Formal Representability and Recursive Enumerability
     III. Incompleteness and Undecidability
     IV. Recursive Function Theory
     V. Creativity and Effective Inseparability
     Supplement: Applications to Mathematical Logic
     Reference and Brief Bibliography
   
[Smullyan1995]
Smullyan, Raymond M.  First-Order Logic.  Dover Publications (New York: 1968, 1995).  ISBN 0-486-68370-2 pbk.  Unabridged, corrected republication of the work first published by Springer-Verlag, New York, 1968.
   
[Solovay1986]
Gödel, Kurt., Feferman, Solomon (editor-in-chief), Dawson, John W. Jr., Kleene, Stephen C., Moore, Gregory H., Solovay, Robert M., van Heijenoort, Jean (eds.).  Kurt Gödel: Collected Works, vol.1: Publications 1929-1936.  Oxford University Press (New York: 1986).  ISBN 0-19-514720-0 pbk.  See [Gödel1986]
    
[Solovay1990]
Gödel, Kurt., Feferman, Solomon (editor-in-chief), Dawson, John W. Jr., Kleene, Stephen C., Moore, Gregory H., Solovay, Robert M., van Heijenoort, Jean (eds.).  Kurt Gödel: Collected Works, vol.2: Publications 1938-1974.  Oxford University Press (New York: 1990).  ISBN 0-19-514721-9 pbk.  See [Gödel1990]
   
[Solovay1995]
Gödel, Kurt., Feferman, Solomon (editor-in-chief)., Dawson, John W. Jr., Goldfarb, Warren., Parsons, Charles., Solovay, Robert M. (eds.).  Kurt Gödel: Collected Works, vol.3: Unpublished essays and lectures.  Oxford University Press (New York: 1995).  ISBN 0-19-514722-7 pbk.  See [Gödel1995]
   
[Stoll1963]
Stoll, Robert R.  Set Theory and Logic.  Dover Publications (New York: 1961, 1964).  ISBN 0-486-63829-4.
     This book subsumes the material of Stoll's earlier Sets, Logic, and Axiomatic Theories, providing more comprehensive treatment.
     "Cantor's investigation of questions pertaining to trigonometric series and series of real numbers led him to recognize the need for a means of comparing the magnitude of infinite sets of numbers.  To cope with this problem, he introduced the notion of the power (or size) of a set by defining two sets as having the same power if the members of one can be paired with those of the other.  ... The notion of power for infinite sets provides a generalization of everyday counting numbers.  Cantor developed the theory, including an arithmetic, of these generalized (or transfinite) numbers and in so doing created a theory of sets.  His accomplishments in this area are regarded as an outstanding example of mathematical creativity.
     "Cantor's insistence on dealing with the infinite as an actuality ... was an innovation at that time.  Prejudices against this viewpoint were responsible for the rejection of his work by some mathematicians, but others reacted favorably because the theory provided a proof of the existence of transcendental numbers.  Other applications in analysis and geometry were found, and Cantor's theory of sets won acceptance to the extent that by 1890 it was recognized as an autonomous branch of mathematics.  About the turn of the century there was some change in attitude with the discovery that contradictions could be derived within the theory.  That these were not regarded as serious defects is suggested by their being called paradoxes--defects which could be resolved, once full understanding was acquired.  ..."  -- from Chapter 1, Sets and Relations, pp. 1-2.
     The development of Set Theory is along the standard Zermelo-Fraenkel lines (cf. [Suppes1972]).  The von Neumann-Bernays-Gödel approach is sketched, with appropriate references to the original formulations.   This treatment provides coverage of Boolean algebras and algebraic systems, going beyond the foundational treatments of other works.  It ends with a discussion of Skolem's paradox and the uncertainty we are left with.  --dh:2002-07-24
   Contents
     Preface
     Chapter 1.  Sets and Relations
          1. Cantor's Concept of a Set
          2. The Basis of Intuitive Set Theory
          3. Inclusion
          4. Operations for Sets
          5. The Algebra of Sets
          6. Relations
          7. Equivalence Relations
          8. Functions
          9. Composition and Inversion for Functions
          10. Ordering Relations
     Chapter 2. The Natural Number Sequence and Its Generalizations
          1. The Natural Number Sequence
          2. Proof and Definition by Induction
          3. Cardinal Numbers
          4. Countable Sets
          5. Cardinal Arithmetic
          6. Order Types
          7. Well-ordered Sets and Ordinal Numbers
          8. The Axiom of Choice, the Well-ordering Theorem, and Zorn's Lemma
          9. Further Properties of Cardinal Numbers
          10. Some Theorems Equivalent to the Axiom of Choice
          11. The Paradoxes of Intuitive Set Theory
     Chapter 3. The Extension of the Natural Numbers to the Real Numbers
          1.  The System of Natural Numbers
          2. Differences
          3. Integers
          4. Rational Numbers
          5. Cauchy Sequences of Natural Numbers
          6. Real Numbers
          7. Further Properties of the Real Number Systems
     Chapter 4. Logic
          1. The Statement Calculus.  Sentential Connectives
          2. The Statement Calculus.  Truth Tables
          3. The Statement Calculus.  Validity
          4. The Statement Calculus.  Consequence
          5. The Statement Calculus.  Applications
          6. The Predicate Calculus.  Symbolizing Everyday Language
          7. The Predicate Calculus.  A Formulation
          8. The Predicate Calculus.  Validity
          9. The Predicate Calculus.  Consequence
     Chapter 5. Informal Axiomatic Mathematics
          1. The Concept of an Axiomatic Theory
          2. Informal Theories
          3. Definitions of Axiomatic Theories by Set-theoretical Predicates
          4. Further Features of Informal Theories
     Chapter 6. Boolean Algebras
          1. A Definition of a Boolean Algebra
          2. Some Basic Properties of a Boolean Algebra
          3. Another Formulation of the Theory
          4. Congruence Relations for a Boolean Algebra
          5. Representations of Boolean Algebras
          6. Statement Calculi as Boolean Algebras
          7. Free Boolean Algebras
          8. Applications of the Theory of Boolean Algebras to Statement Calculi
          9. Further Interconnections between Boolean Algebras and Statement Calculi
     Chapter 7. Informal Axiomatic Set Theory
          1. The Axioms of Extension and Set Formation
          2. The Axioms of Pairing
          3. The Axioms of Union and Power Set
          4. The Axioms of Infinity
          5. The Axiom of Choice
          6. The Axiom Schemas of Replacement and Restriction
          7. Ordinal Numbers
          8. Ordinal Arithmetic
          9. Cardinal Numbers and Their Arithmetic
          10. The von Neumann-Bernays-Gödel Theory of Sets
     Chapter 8. Several Algebraic Theories
          1. Features of Algebraic Theories
          2. Definition of a Semigroup
          3. Definition of a Group
          4. Subgroups
          5. Coset Decompositions and Congruence Relations for Groups
          6. Rings, Integral Domains, and Fields
          7. Subrings and Difference Rings
          8. A Characterization of the System of Integers
          9. A Characterization of the System of Rational Numbers
          10. A Characterization of the Real Number System
     Chapter 9.  First-Order Theories
          1. Formal Axiomatic Theories
          2. The Statement Calculus as a Formal Axiomatic Theory
          3. Predicate Calculi of First Order as Formal Axiomatic Theories
          4. First-order Axiomatic Theories
          5. Metamathematics
          6. Consistency and Satisfiability of Sets of Formulas
          7. Consistency, Completeness, and Categoricity of First-order Theories
          8. Turing Machines and Recursive Functions
          9. Some Undecidable and Some Decidable Theories
          10. Gödel's Theorems
          11. Some Further Remarks about Set Theory
     References
     Symbols and Notation
     Author Index
     Subject Index
   
[Stolyar1970]
Stolyar, Abram Aronovich.  Introduction to Elementary Mathematical Logic.  Dover (New York: 1970).  ISBN 0-486-64561-4 pbk.  Unabridged and unaltered 1983 republication of the work published by MIT Press (Cambridge, MA: 1970).  Translation of Elementarnoe vvedenie v matematicheskuiu logiku, Prosveshcheniye Press (Moscow: 1965), with translation from the Russian edited by Elliot Mendelson.
     "By virtue of its numerous applications in the most varied fields of science and technology ..., present-day mathematical logic is attracting the attention of a large number of persons in various specialties, including high-school mathematics teachers.
     "The elements of mathematical logic and certain of its applications are included in the programs of high schools that place emphasis on mathematics.  They can serve as interesting material for out-of-class work for students in the higher grades of any high school. ...
     "... This book cannot be regarded as an introduction to mathematical logic as a whole.  However, a familiarity with the material expounded in it will make it easier for the reader who is seriously interested in studying the subject to read the literature recommended for this purpose at the end of the book."  -- From the Author's Preface.
     This book was developed for high school students.  I recommend it as an introduction to logic because of its availability (under $10 in 2002), gentle pace, and introduction to the formalization of logic with careful attention to motivation and application.  In the past, access to elementary formal/axiomatic logic arose in the works designed to popularize the symbology and approach of Principia Mathematica.  Stolyar provides a treatment that is consistent with the now-popular notations of formalized theories, carrying the reader through to first-order logic with equality.  This is a nice starter kit for further exploration and the application of logic in computer science, say by studying Enderton.  -- dh:2002-07-25.
   Content
     Author's Preface
     Introduction
     Chapter 1: Propositional Logic
          1. Objects and operations
          2. Formulas.  Equivalent formulas.  Tautologies.
          3. Examples of the application of the laws of the logic of propositions in derivations.
          4. Normal forms of functions.  Minimal forms.
          5. Application of the algebra of propositions to the synthesis and analysis of discrete-action networks.
     Chapter 2: The Propositional Calculus
          1. The axiomatic method.  The construction of formalized languages.
          2. Construction of a propositional calculus (alphabet, formulas, derived formulas).
          3. Consistency, independence, and completeness of a system of axioms in the propositional calculus.
     Chapter 3: Predicate Logic
          1. Sets.  Operations on sets.
          2. The inadequacy of propositional logic.  Predicates.
          3. Operations on predicates.  Quantifiers.
          4. Formulas of predicate logic.  Equivalent formulas.  Universally valid formulas.
          5. Traditional logic (the logic of one-place predicates).
          6. Predicate logic with equality.  Axiomatic construction of mathematical theories in the language of predicate logic with equality.
     Appendices
          I. A proof of the duality principle for propositional logic.
          II. A proof of the deduction theorem for the propositional calculus.
          III. A proof of the completeness theorem for the propositional calculus.
     Bibliography
     Index of Special Symbols
     Index
   
[Suppes1972]
Suppes, Patrick Colonel.  Axiomatic Set Theory.  D. Van Nostrand (New York: 1960).  Unabridged and corrected republication with new preface and section 8.4, Dover Publications (New York: 1972).  ISBN 0-486-61630-4 pbk.
     "This book is intended primarily to serve as a textbook for courses in axiomatic set theory.  The Zermelo-Fraenkel system is developed in detail.  The mathematical prerequisites are minimal; in particular, no previous knowledge of set theory or mathematical logic is assumed.  On the other hand, students will need a certain degree of general mathematical sophistication, especially to master the last two chapters.  Although some logical notations is used throughout the book, proofs are written in an informal style and an attempt has been made to avoid excessive symbolism.  A glossary of the more frequently used symbols is provided." -- from the Preface to the First Edition, p.vii.
     "The working mathematician, as well as the man in the street, is seldom concerned with the unusual question: What is a number?  But the attempt to answer this question precisely has motivated much of the work by mathematicians and philosophers in the foundations of mathematics during the past hundred years.  Characterization of the integers, rational numbers and real numbers has been a central problem for the classical researches of Weierstrass, Dedekind, Kronecker, Frege, Peano, Russell, Whitehead, Brouwer, and others.  ...
     "Yet the real development of set theory was not generated directly by an attempt to answer this central problem of the nature of number, but by the researches of Georg Cantor around 1870 in the theory of infinite series and related topics of analysis.  Cantor, who is usually considered the founder of set theory as a mathematical discipline, was led by his work into a consideration of infinite sets or classes of arbitrary character.  In 1874 he published his famous proof that the set of real numbers cannot be put into one-one correspondence with the set of natural numbers (the non-negative integers).  In 1878 he introduced the fundamental notion of two sets being equipollent or having the same power (Mächtigkeit) if they can be put into one-one correspondence with each other.  ... The notion of power leads in the case of infinite sets to a generalization of the notion of a natural number to that of an infinite cardinal number.  Development of the general theory of transfinite numbers was one of the great accomplishments of Cantor's mathematical researches. 
     " ... From the standpoint of the foundations of mathematics the philosophically revolutionary aspect of Cantor's work was his bold insistence on the actual infinite, that is, on the existence of infinite sets as mathematical objects on a par with numbers and finite sets.  There is scarcely a serious philosopher of mathematics since Aristotle who has not been much exercised about this difficult concept." -- from the Introduction §1.1, pp.1-2.
     I have been mining in here to train myself at clarifying questions involving denumerable sets as the ceiling under which computation (and any other effective process) is compelled to operate.  This leads me to take a cycle around the foundational topics of set, number, and the transfinite.  In selecting sections to identify the expanded content for, I realized that the only ones I might safely neglect are chapters 3 and 7, and I am not so certain of that.  So here are all of the section titles.  -- dh:2002-07-24
   Content
     Preface to the Dover Edition
     Preface to the First Edition
     1. Introduction
          1.1 Set Theory and the Foundations of Mathematics
          1.2 Logic and Notation
          1.3 Axiom Schema of Abstraction and Russell's Paradox
          1.4 More Paradoxes
          1.5 Preview of Axioms
     2. General Developments
          2.1 Preliminaries: Formulas and Definitions
          2.2 Axioms of Extensionality and Separation
          2.3 Intersection, Union, and Difference of Sets
          2.4 Pairing Axiom and Ordered Pairs
          2.5 Definition by Abstraction
          2.6 Sum Axiom and Families of Sets
          2.7 Power Set Axiom
          2.8 Cartesian Product of Sets
          2.9 Axiom of Regularity
          2.10 Summary of Axioms
     3. Relations and Functions
          3.1 Operations on Binary Relations
          3.2 Ordering Relations
          3.3 Equivalence Relations and Partitions
          3.4 Functions
     4. Equipollence, Finite Sets, and Cardinal Numbers
          4.1 Equipollence
          4.2 Finite Sets
          4.3 Cardinal Numbers
          4.4 Finite Cardinals
     5. Finite Ordinals and Denumerable Sets
          5.1 Definition and General Properties of Ordinals
          5.2 Finite Ordinals and Recursive Definitions
          5.3 Denumerable Sets
     6. Rational Numbers and Real Numbers
          6.1 Introduction
          6.2 Fractions
          6.3 Non-negative Rational Numbers
          6.4 Rational Numbers
          6.5 Cauchy Sequences of Rational Numbers
          6.6 Real Numbers
          6.7 Sets of the Power of the Continuum
     7. Transfinite Induction and Ordinal Arithmetic
          7.1 Transfinite Induction and Definition by Transfinite Induction
          7.2 Elements of Ordinal Arithmetic
          7.3 Cardinal Numbers Again and Aleph
          7.4 Well-Ordered Sets
          7.5 Revised Summary of Axioms
     8. The Axiom of Choice
          8.1 Some Applications of the Axiom of Choice
          8.2 Equivalents of the Axiom of Choice
          8.3 Axioms Which Imply the Axiom of Choice
          8.4 Independence of the Axiom of Choice and the Generalized Continuum Hypothesis
     References
     Glossary of Symbols
     Author Index
     Subject Index
   
[Tarski1953]
Tarski, Alfred., Mostowski, Andrzej., Robinson, Raphael M.  Undecidable Theories.  North-Holland (Amsterdam: 1953).  Studies in Logic and the Foundation of Mathematics.
   Content
     Preface
     I. A General Method in Proofs of Undecidability, Alfred Tarski
          I.1 Introduction
          I.2 Theories with standard formulation
          I.3 Undecidable and essentially undecidable theorems
          I.4 Interpretability and weak interpretability
          I.5 Relativization of quantifiers
          I.6 Examples and applications
     II. Undecidability and Essential Undecidability in Arithmetic, Andrzej Mostowski, Raphael M. Robinson, and Alfred Tarski
          II.1 A summary of results; notations
          II.2 Definability in arbitrary theories
          II.3 Formalized arithmetic of natural numbers and its sub-theories
          II.4 Recursiveness and definability in subtheories of arithmetic
          II.5 Undecidability of subtheories of arithmetic
          II.6 Extension of the results to other arithmetical theories and to various theories of rings
     III. Undecidability of the Elementary Theory of Groups, Alfred Tarski
     Bibliography
     Index
   
[Tarski1956]
Tarski, Alfred.  Logic, Semantics, Metamathematics: Papers from 1923 to 1938. translated by J. H. Woodger.  Oxford University Press (London: 1956).
     This book is in the holdings of the Seattle Public Library and I needed a member of the staff to check it out to me.  It is also not renewable and I am capturing the contents before returning it unread (this time).  Some of the articles have been updated slightly from the original published versions, so this is generally a superior reference to many of those earlier papers.  dh:2004-10-22.
   Content
     Translator's Preface
     Author's Acknowledgments
     1. On the Primitive Term of Logistic [1923]
     2. Foundations of the Geometry of Solids [1929]
     3. On Some Fundamental Concepts of Metamathematics [1930]
     4. Investigations into the Sentential Calculus (with Jan Łukasiewicz) [1930]
     5. Fundamental Concepts of the Methodology of the Deductive Sciences [1930]
     6. On Definable Sets of Real Numbers [1931]
     7. Logical Operations and Projective Sets (with Casimir Kuratowski) [1931]
     8. The Concept of Truth in Formalized Languages [1935]
     9. Some Observations on the Concepts of ω-Consistency and ω-Completeness [1933]
     10. Some Methodological Investigations on the Definability of Concepts [1935]
     11. On the Foundations of Boolean Algebra [1935]
     12. Foundations of the Calculus of Systems [1936]
     13. On the Limitations of the Means of Expression of Deductive Theories (with Adolf Lindenbaum) [1935]
     14. On Extensions of Incomplete Systems of the Sentential Calculus [1935]
     15. The Establishment of Scientific Semantics [1936]
     16. On the Concept of Logical Consequence [1936]
     17. Sentential Calculus and Topology [1938]
     Abbreviations
     Bibliography
     Subject Index
     Index of Names of Persons
     Index of Symbols
[vanHeijenoort1977]
van Heijenoort, Jean (ed).  From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931.  Harvard University Press (Cambridge, MA: 1967), 3rd (1977) printing.  ISBN 0-674-32449-8 (paper).
   Contents
     General Editor's Preface
     Preface
     Note to the Second Printing
     Note to the Third Printing
     Gottlob Frege.  Begriffsschrift, a formal language, modeled upon that of arithmetic, for pure thought [1879]
     Giuseppe Peano.  The principles of arithmetic, presented by a new method [1889]
     Richard Dedekind.  Letter to Keferstein [1890a]
     Cesare Burali-Forti.  A question on transfinite numbers [1897]
     Cesare Burali-Forti. On well-ordered classes [1897a]
     Georg Cantor.  Letter to Dedekind [1899]
     Alessandro Padoa.  Logical introduction to any deductive theory [1900]
     Bertrand Russell.  Letter to Frege [1902]
     Gottlob Frege.  Letter to Russell [1902]
     David Hilbert.  On the foundations of logic and arithmetic [1904]
     Ernst Zermelo.  Proof that every set can be well-ordered [1904]
     Jules Richard.  The principles of mathematics and the problem of sets [1905]
     Julius König.  On the foundations of set theory and the continuum problem [1905a]
     Bertrand Russell.  Mathematical logic as based on the theory of types [1908a]
     Ernst Zermelo.  A new proof of the possibility of a well-ordering [1908]
     Ernst Zermelo.  Investigations in the foundations of set theory I [1908a]
     Alfred North Whitehead and Bertrand Russell.  Incomplete symbols: Descriptions [1910]
     Norbert Wiener.  A simplification of the logic of relations [1914]
     Leopold Löwenheim.  On possibilities in the calculus of relatives [1915]
     Thoralf Skolem.  Logico-combinatorial investigations in the satisfiability or provability of mathematical propositions: A simplified proof of a theorem by L. Löwenheim and generalizations of the theorem [1920]
     Emil Leon Post.  Introduction to a general theory of elementary propositions [1921]
     Abraham A. Fraenkel.  The notion "definite" and the independence of the axiom of choice [1922b]
     Thoralf Skolem.  Some remarks on axiomatized set theory [1922]
     Thoralf Skolem.  The foundations of elementary arithmetic established by means of the recursive mode of thought, without the use of apparent variable ranging over infinite domains [1923]
     Luitzen Egbertus Jan Brouwer.  On the significance of the principle of excluded middle in mathematics, especially in function theory. [1923b, 1954, 1954a]
     John von Neumann.  On the introduction of transfinite numbers.  [1923]
     Moses Schönfinkel.  On the building blocks of mathematical logic [1924]
     David Hilbert.  On the infinite [1925]
     John von Neumann.  An axiomatization of set theory [1925]
     Andrei Nikolaevich Kolmogorov.  On the principle of excluded middle [1925]
     Paul Finsler.  Formal proofs and undecidability [1926]
     Luitzen Egbertus Jan Brouwer.  On the domains of definition of functions [1927]
     David Hilbert.  The foundations of mathematics [1927]
     Hermann Weyl.  Comments on Hilbert's second lecture on the foundations of mathematics [1927]
     Paul Bernays.  Appendix to Hilbert's lecture "The foundations of mathematics" [1927a]
     Luitzen Egbertus Jan Brouwer.  Intuitionistic reflections on formalism [1927]
     William Ackermann.  On Hilbert's construction of the real numbers [1928]
     Thoralf Skolem.  On mathematical logic [1928]
     Jacques Herbrand.  Investigations in proof theory: The properties of true propositions [1930]
     Kurt Gödel.  The completeness of the axioms of the functional calculus of logic [1930a]
     Kurt Gödel.  Some metamathematical results on completeness and consistency [1930b]
     Kurt Gödel.  On formally undecidable propositions of Principia Mathematica and related systems [1931]
     Kurt Gödel.  On completeness and consistency [1931a]
     Jacques Herbrand.  On the consistency of arithmetic [1931]
     References
     Index
     Corrections made in the second printing
     Corrections made in the third printing
   
[vanHeijenoort1986]
Gödel, Kurt., Feferman, Solomon (editor-in-chief), Dawson, John W. Jr., Kleene, Stephen C., Moore, Gregory H., Solovay, Robert M., van Heijenoort, Jean (eds.).  Kurt Gödel: Collected Works, vol.1: Publications 1929-1936.  Oxford University Press (New York: 1986).  ISBN 0-19-514720-0 pbk.  See [Gödel1986]
   
[vanHeijenoort1990]
Gödel, Kurt., Feferman, Solomon (editor-in-chief), Dawson, John W. Jr., Kleene, Stephen C., Moore, Gregory H., Solovay, Robert M., van Heijenoort, Jean (eds.).  Kurt Gödel: Collected Works, vol.2: Publications 1938-1974.  Oxford University Press (New York: 1990).  ISBN 0-19-514721-9 pbk.  See [Gödel1990]
   
[vonNeumann1923]
von Neumann, John.  Zur Einführung der transfiniten Zahlen.  Acta litterarum ac scientiarum Regiae Universitatis Hungaricae Francisco-Josephinae, Section scientiarum mathematicarum 1 (1923), 199-208.  Translated with an editorial preface by Jean van Heijenoort as "On the introduction of transfinite numbers" on pp. 346-354 in From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931.  Jean van Heijenoort, editor.  Harvard University Press (Cambridge, MA: 1967), 3rd (1977) printing.  ISBN 0-674-32449-8 (paper).  See [vanHeijenoort1977]
   
[vonNeumann1925]
von Neumann, John.  Eine Axiomatisierung der Mengenlehre (An Axiomatization of Set Theory).  Journal für die reine und angewandte Mathematik 154, 219-240.  Berichtigung, ibid. 155, 128.  Translated by Stefan Bauer-Mengelberg and Dagfinn Fellesdal with an editorial preface as "An axiomatization of set theory" on pp. 393-413 in From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931.  Jean van Heijenoort, editor.  Harvard University Press (Cambridge, MA: 1967), 3rd (1977) printing.  ISBN 0-674-32449-8 (paper).  See [vanHeijenoort1977]
   
[Whitehead1997]
Whitehead, Alfred North., Russell, Bertrand.  Principia Mathematica to *56.  Cambridge Mathematical Library edition.  Cambridge University Press (London: 1910, 1927, 1962, 1997).  ISBN 0-521-62606-4 pbk.
   Contents
     Preface (1910)
     Alphabetical List of Propositions Referred to by Names
     Introduction to the Second Edition (1927)
     Introduction
          I. Preliminary Explanations of Ideas and Notations
          II. The Theory of Logical Types
          III. Incomplete Symbols
     Part I. Mathematical Logic
          A. The Theory of Deduction
          B. Theory of Apparent Variables
          C. Classes and Relations
          D. Logic of Relations
     Part II. Prolegomena to Cardinal Arithmetic
          A. Unit Classes and Couples
     Appendix A
          *8.  The Theory of Deduction for Propositions containing Apparent Variables
     Appendix C
          Truth-Functions and others
     List of Definitions
   
[Wikipedia2018b]
Structural induction. 2018-09-09 article available on the Internet at <https://en.wikipedia.org/w/index.php?title=Structural_induction&oldid=858748480>.
  
[Wikipedia2019]
First-order logic. 2019-01-07 article available on the Internet at <https://en.wikipedia.org/w/index.php?title=First-order_logic&oldid=877262034>
  
[Zermelo1908a]
Zermelo, Ernst.  Untersuchungen über die Grundlagen der Mengenlehre I (Investigations in the Foundations of Set Theory).  Mathematische Annalen 65 (1908), 261-281.  Translation by Stefan Bauer-Mengelberg with introductory note by Jean van Heijenoort as "Investigations in the foundations of set theory I" on pp. 199-215 in From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931.  Jean van Heijenoort, editor.  Harvard University Press (Cambridge, MA: 1967), 3rd (1977) printing.  ISBN 0-674-32449-8 pbk.  See [vanHeijenoort1977]
     This is the paper in which Zermelo sets out to the first axiomatic system for Cantor's set theory.  With Abraham Fraenkel's improvement, it has remained intact in every fundamental respect as the ever-popular system, ZF, or its cousin, ZFC (ZF + Axiom of Choice).  The differences between this progression and the alternative formulation, [von Neumann-]Bernays-Gödel, do not impact the standing of the transfinite and of the non-too-large cardinals.
     The paper is divided into two sections, §1 Fundamental Definitions and Axioms and §2 Theory of Equivalence, the treatment of cardinality and the transfinite. -- dh:2002-07-25
  
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