[FOM] explicit construction; choice and model theory
John T. Baldwin
jbaldwin at uic.edu
Tue Jul 1 13:11:38 EDT 2003
This is a response to one of Harvey's lines arising from my post:
CATEGORICITY and STRONGLY MINIMAL SETS
>
> Friedman wrote:
>
>>>
>>> Let T be a first order sentence that has an infinite model. I have
>>> been interested in the question of whether you can explicitly
>>> construct a model of T whose domain is a given infinite set D.
>>
Now I understand what seems to me a very strange question. Here is
the obvious response.
INSTINCTUAL RESPONSE: Model theory is based on a basically
`structuralist' attitude. A model is a set
D and a collection of relations and functions. Any bijection between D
and D' carries over to an isomorphism of the stuctures.
Whether a bijection exists between two sets is a set theoretic not a
model theoretic question.
2 second thoughts.
1) There are a few important exceptions to general statement that the
structure of the domain is not relevant to model theory.
More famous is Shelah's solution to the Whitehead problem which uses
explictly the fact (choice) that an Abelian group of cardinality aleph_1
can be taken to have aleph_1 as its domain and then uses properties of
aleph_1 essentially in the proof.
Of more contemporary interest, many results in the model theory of
infinitary logic use a similar concrete representation and require
some weak diamond like principles.
In another direction, Philip Ehrlich in the JSL a few years ago
extablishes results about real closed field using an explicit represenation
in terms of the fields as objects in Conway's surreal numbers.
Now a foundational problem might be to try to sort our what kinds of
problems really require this concrete representation of the
model.
2) Work in ZF. Call two sets equivalent if there is a bijection
between them. Apparently, Harvey is telling me that the
Lowenheim-Skolem-Tarski theorem has a complicated formulation here.
Without choice, he says it is impossible to prove that every
sentence has a model in each equivalence class (with infinite elements).
Sounds like a good argument for working in ZFC :)
Later Friedman writes:
I would think that behind every set theoretically formulated theorem in
model theory that model theorists of the modern kind really care about,
there is a formulation that is not very set theoretic. In particular,
well orderings should play NO ROLE for the modern model theorist.
Baldwin replies. Harvey, I think you have a strange notion of `modern
model theorist'. Perhaps you read too much Macintyre.
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