Bibliography Readings/Notes |
These are notes based on consideration of Church, Alonzo. Introduction to Mathematical Logic. Princeton University Press (Princeton, NJ: 1944, 1956). ISBN 0-691-02906-7 pbk. With 1958 errata..
Alonzo Church (1903 - 1995) was a longtime member of the Princeton University department of mathematics. Thereafter he was a professor of philosophy and mathematics at UCLA. Alan Turing was one of his graduate students, as were several other now well-known logicians and philosophers.
There is much here to consider with regard to models. I think I didn't quite have it in 2000. I need to re-assess it with regard to the formulation of the structure, ‹ob›. This page will remain for reference purposes. I don't foresee developing it further in this form.
-- Dennis E. Hamilton
Seattle, Washington
2014-05-03
I have been saying that Miser is a model, and I now need to be rigorous about it. And I would like to keep as much harmony with the use of the term in mathematical logic and also in science.
Using the index, we have a mention in footnote 451 on p.287 (in §48.22). This relates to a system, G, having enumerably infinite models, and those having enumerably infinite extensions.
In §55, on postulate theory, Church takes this apart.
When a system of postulates is used as basis for the formal treatment of some mathematical theory, the postulates have to be thought of as added to an underlying logic.
Viz., for Miser as a theory, there is an underlying logic to which the particular postulates and definitions are added.For the precise syntactical definition of the particular theory, it is necessary to state not only the specific mathematical postulates but also a formalization of the underlying logic.
The class of theorems belonging to the theory in question is determined by both the postulates and the underlying logic.
Church looks at the interplay of these. He is careful in forming theories to not use constants but to have uninterpreted postulates. He will then speak of interpretations of the theory (e.g., for arithmetic and for a number theory (n-th order arithmetic).
Church considers that there is a representing form of postulates in which it is expressed as a universal closure (that is, with no free variables, or at least no free variables of a certain kind), and it is added to a functional calculus (defined earlier) of the least order in which all of the representing forms are well-formed.
A model of the postulates, in this case, is a non-empty domain of individuals together with a system of values of the free variables of the representing form which satisfy the postulates simultaneously in the domain of individuals.
We now get to a very interesting place. Consider a propositional form in the system. This form is a consequence of the postulates if it is true in every model of the postulates.
The mathematical theory can be considered to be all of the propositions (within the system) that are consequences of the postulates. [This inherently includes the postulates themselves.]
Two models of a system of postulates are said to be isomorphic if there is a one-to-one correspondence between the two domains of individuals of the models, such that the assignments of free variables always correspond between the two models.
A system of postulates is considered categorical if all of its models are isomorphic.
It looks like what I have to deal with here is a confluence of system, theory, model, and interpretation.
Church's sense of interpretation is that used now in mathematical logic. It would appear that his sense of model also agrees. I need to now look at other sources to determine how to reconcile my sense of theories as modeling something versus theories having models.
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