[FOM] explicit construction; choice and model theory
Harvey Friedman
friedman at math.ohio-state.edu
Tue Jul 1 18:12:02 EDT 2003
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.
>
>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.
>
>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?
>
>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.
>
>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.
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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