[FOM] explicit construction; choice and model theory

[John T. Baldwin]

I make a couple of comments below.



Harvey Friedman wrote:

> Reply to Baldwin 12:11PM 7/1/03.
>
> This interchange is really about what model theory might look like if 
> one pays systematic attention to certain foundational matters 
> surrounding explicitness.
>
> My thinking is that this point of view is not at odds with the current 
> trends I see in papers and meetings in current model theory.
>
> NOTE: This foundational interchange should not slow down any plans for 
> your presentations of model theoretic material.
>
>>>
>>>>>
>>>>> 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.
>
>
> You are a distinguished model theorist, and your "instincts" do not 
> look very foundational to me. I don't know if there is a cause or 
> effect here, in either or both directions (smile).
>
> The mainstream of mathematical culture is with the finite and the 
> countable. This includes Borel functions between Polish spaces at the 
> outer boundary, as it is clearly understood in countable terms.
>
> So one Corollary of this viewpoint is to look for finite and countable 
> formulations of everything in model theory. In fact, one 
> interpretation of what a lot of current model theorists do is just 
> that - at least that's my impression from listening to MacIntyre, van 
> den Dries, Zilber, maybe Hrushovski. MacIntyre is on the FOM.
>
> Of course, I'm not past showing that this mainstream is, quite 
> surprisingly, NOT  self contained. I make a living off of that 
> (smile). In fact, I want to show that even working with structures 
> whose domain has 100 elements cannot be handled properly without the 
> largest of the large cardinals. But that's a different, non model 
> theoretic (at least now) foundational story.

This is responsive to my comments below. It is not responsive to the 
basic point that the set theoretic structure of the domain is just NOT 
model theory.  I
think that Angus and Lou would agree with me on this.

>
>>
>> 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.
>
>
> I call that work set theory, or set theoretic algebra. This is not at 
> all what I have in mind. 

Ah, but it is a use of model theory to solve real mathematical problems. 
 By require below  I meant could these model theoretic/ algebra problems
(which do not apparently depend on the representation of the domain as 
an ordinal) be solved without reference to that concrete representation.  

>
>
>>
>> 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.
>
>
> What kind of infinitary logic? The kind most clearly related to 
> foundations of mathematics is countably infinitary logic, which merges 
> with descriptive set theory, and Borel structures with D = the reals 
> or a Polish space, is what is relevant here. Again, a great deal of 
> explicitness, and no well orderings! Recall Borel Model Theory - 
> perhaps a dead subject (I set up, if I recall), worth pulling up out 
> of the grave? 

I mean questions like whether a sentence in L omega one  which is 
aleph_1 categorical has model of power aleph_2.

>
>
>>
>> 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.
>
>
> Which real closed field? There are explicit ones and nonexplicit ones. 

check the paper

>
>
>>
>> Now a foundational problem might be to try to sort our what kinds of 
>> problems really require this concrete representation of the
>> model.
>
>
> "Require" is not how I would say this. It is an end in itself, an 
> integral part of paying attention to the foundational landscape of 
> actual mathematics.
>
> However, see below, where in a sense, one does have a kind of indirect 
> requirement of this kind.
>
>>
>> 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  :)
>
>
> There is nothing complicated about this. It illustrates very strongly 
> the special importance of injective binary functions and linear 
> orderings at the very beginnings of model theory, in a highly 
> nontechnical context. Something that comes up much later in the 
> development of the subject (although still part of basic material, of 
> course).
>
> So suppose you want to illustrate the special status of injective 
> binary functions and linear orderings in the most elemental way for 
> model theory. How do you do this? You ask for an explicit form of the 
> Upward Lowenheim Skolem Theorem. Then it is forced on you. "Required", 
> as you say. 

With AC, we have a sentence which has infinite models has one in every 
cardinal?

What is the ZF statement?  If read you right it is:  There is a model 
with domain X if there is a bijection of X with a D ssuch that
D is infinite, there is a one-one binary function from D into D, and 
there is a linear ordering of D.

It seems like a more complicated way to say some that is fundamentally 
irrelevant - because  model theory (this is now a mantra)
is not concerned with structure of the domain.


>
>
> I doubt very much if this explicit approach would in any way force 
> model theorists to take turns that they don't already take in some 
> sense, or that they wouldn't want to take.
>
> I.e., look for explicitness and explicit formulations, when they are 
> missing, and do so systematically.
>
>>
>> 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.
>
>
> This way of looking at things is also very much in the mind of van den 
> Dries and his students - in fact, often too extreme.
>
>>
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