[FOM] explicit construction; choice and model theory

[Harvey Friedman]

Reply to Baldwin 7:32PM  7/2/03.

It appears that Baldwin and I are looking at model theory from quite 
different perspectives. It is probably productive and of interest to 
the FOM to discuss this in detail.

Let me go into this a little bit before I answer Baldwin's posting directly.

FRIEDMAN'S VIEW OF MODEL THEORY.

Contemporary model theory is a branch of mathematical logic, which 
like the other major branches (recursion theory, set theory, proof 
theory) - grew out of fundamental work in the foundations of 
mathematics. Because model theory has turned out to be perhaps 
unexpectedly useful in core mathematics, model theorists now have an 
eye towards finding more applications to core mathematics, and this 
has definitely changed the structure of the field from what it would 
be otherwise. In fact, there has been an added investment made in 
contexts where the hope of applications to core mathematics looks 
most promising.

Since core mathematics is entirely concerned with rather concrete and 
explicit situations, the "hottest" model theory now concentrates on 
such concrete and explicit situations.

Putting aside the obvious motivating force of applications to core 
mathematics, model theory can be viewed, just like recursion theory 
and set theory and proof theory, as a focused branch of mathematics 
that naturally comes out of previous great work in the foundations of 
mathematics.

Of course, the fact that model theory - and the other main branches 
of mathematical logic - came out of previous great work in the 
foundations of mathematics, does NOT mean that model theory uses 
foundations of mathematics for motivation, nor does it mean that 
model theory is or is not of significance for the foundations of 
mathematics.

 From this perspective, Friedman evaluates work in model theory in 
terms of what insight it does - or can be made to do - for 
foundations of mathematics.

This previous paragraph is a Corollary to Friedman's evaluation of 
all intellectual activity - in terms of its general intellectual 
interest. Foundations of mathematics is the study of features of 
mathematics - normally epistemological and/or ontological - of 
general intellectual interest.

Applications of model theory to core mathematics do not, in and of 
themselves, provide any insight into the foundations of mathematics. 
Of course, it may provide such insight, but normally only if 
additional work is done to connect up with foundations of mathematics.

As a Corollary to the above, Friedman will always seek concrete and 
explicit formulations of any mathematical work whatsoever, including 
model theory.

So when Friedman is confronted with the Upward Lowenheim Skolem 
Theorem, Friedman will look to a concrete and explicit form of it.

BALDWIN'S VIEW OF MODEL THEORY.

(Guess).

Baldwin views model theory as a focused mathematical subject that 
studies the definable sets and relations in arbitrary relational 
structures. Any issues about concreteness or explicitness are not 
relevant. What functions and sets are to be considered is explicitly 
not within the province of model theory.

Sometimes model theory is useful in core mathematics and sometimes it 
is not. But model theory is just as intrinsically important as most 
other focused mathematical subjects, and needs no justification in 
terms of "general intellectual interest" and the like. Nor does it 
need any justification in terms of what it can do for foundations of 
mathematics. As such, it is no different  than other focused 
mathematical subjects.

..........???

>
>Harvey Friedman wrote:
>
>>
>>
>>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.

Baldwin wrote:

>
>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.

If I said that the domain was finite, would that be model theory?

>
>>
>>>
>>>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.

The mathematicians now say, after looking at this kind of work on 
Abelian groups, that "it is not what they had in mind", and do not 
regard it, in retrospect, as addressing any "real" mathematical 
problem. The reason is precisely because the wild Abelian groups that 
can come up do NOT have a concrete representation. If they did, then 
the mathematicians would be interested. Mathematicians are happy if 
there is no concrete representation BUT things are going smoothly - 
however if there is no concrete representation AND things are going 
badly (i.e., independence from ZFC, etc.), then they point to their 
dislike of nonconcreteness and nonexplicitness.

>>
>>
>>>
>>>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.

This is typical of the nonconcrete/nonexplicit brand of model theory, 
very different from Borel Model Theory.

>>
>>
>>>
>>>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?

THEOREM. ZFC. Let D be a set. The following are equivalent.
i) Every sentence with an infinite model has a model with domain D.
ii) D is infinite.

>
>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 such 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.

THEOREM. ZF. Let D be a set. The following are equivalent.
i) Every sentence with an infinite model has a model with domain D.
ii) D is infinite, there is a one-one map from DxD into D, and D is 
linearly ordered.

There is nothing complicated about this, and it reveals a completely 
fundamental phenomenon in a particularly simple way.

I can give a formulation of this that MIGHT come within what you are 
calling model theory. This alternative formulation still fits 
squarely into my perspective.

DEFINITION. A model theoretic site consists of a nonempty set D 
together with a collection S of relations of several variables on D 
which is closed under first order definability. I.e., any relation 
that can be defined from finitely many relations in S, lies in S.

PROBLEM. Let (D,S) be a model theoretic site. Give a necessary and 
sufficient condition for the following to hold. Every sentence with 
an infinite model has a model with domain D whose relations (and 
graphs of its functions) lie in S.

ANSWER???? If and only if D is infinite, S has (the graph of) a 
one-one map from DxD into D, and S has a linear ordering of D.

ANSWER. If and only if S has a model of (a weak fragment of) PA.

PROBLEM. Give a simple direct mathematical answer, not in terms of a 
formal system like PA.

Harvey Friedman