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Topological spaces are an example of a class that is not an elementary class in model theory, i.e.if you believe that all of mathematics is concerned with “mathematical structures” in the sense of model theory, then you run into some trouble, I suppose.
This is an issue about which various expositions, pamphletes etc must have been written. Could you point me to some?
What precisely characterises categories obtained in model theory?
I don’t think there is yet a precise characterisation, but Rosický has done some work in that direction. See e.g. this paper.
Thanks. While I am reading: give me a pamphlete, suppose I claim all of mathematics is about first order structures in the sense of model theory, what do you reply?
What’s worse, consider homotopy types…
Am on the phone now, but just briefly: that text which Zhen Lin points to above – section 5 – is very useful. This should be cited at structure in model theory and maybe related entries.
To say all of mathematics is about first-order structures is naïve, really. It’s not even true of undergraduate mathematics: $\mathbb{N}$ and $\mathbb{R}$ are second-order structures, topological spaces are third-order, $\sigma$-algebras are not finitary…
David, I guess so, but I have very little feeling for model theory. Give me more precise statements. What is absolutely out of the reach of model theory?
I understand it has very strong ties to algebraic geometry. Do the ties carry over to homotopical aspects such as sheaf cohomology etc?
Zhen Lin, hm, maybe I am using “first order” in the wrong way here.
I am thinking of structures as for instance right on the first pages of Hodges‘ “A shorter model theory”. If we stick to that concept, how far do we get? In modern geometry? Foundations of physics?
You can’t even get topological spaces, never mind manifolds or schemes. But you can get Kan complexes, and if we allow so-called “imaginary elements”, you can get their homotopy groups as well. Fibrant replacement is probably not finitary, but it is definable in $L_{\omega_1, \omega}$.
The intersection of model theory and algebraic geometry is more-or-less orthogonal to scheme theory. The methods used are very different, e.g. o-minimal structures, saturated models. I admit I am not familiar with that stuff, but my impression is that they only work with classical varieties over real closed fields and algebraically closed fields.
Thanks! That’s useful.
@Urs,
I don’t know any concrete results, but the recent blustering on the fom mailing list about homotopy type theory made me think a little bit how one would make a formal set-theoretic definition of a homotopy type. I think one needs, even in ZFC, something like ’Scott’s trick’, which replaces an equivalence class that is a proper class by a set, since a homotopy type ’is’ an equivalence class of ’spaces’ (be they Kan complexes or CW complexes) which unless some other method is used, is not a set. (Scott’s trick uses the concept of ordinal rank, which is completely irrelevant for homotopy theory, hence very unnatural).
This isn’t model theory, but I can’t imagine how it would be any less fraught with difficulty.
I have edited structure in model theory a bit, following the above. (Announced also in the corresponding thread).
One more question:
what’s the relation between algebraic category and categories of structures in model theory? It seems from some of the discussion I have seen that these are supposed to be related, but I am not sure.
Every algebraic category whose forgetful functor preserves filtered colimits is the category of models for some first-order theory. The converse is false, even for one-sorted theories.
Topological spaces are an example of a class that is not an elementary class in model theory
The elementary classes are not the only interesting objects in model theory. They have vast generalizations like abstract elementary classes, metric abstract elementary classes etc. But model theory considers also things which have no relation to “elementary” in any sense.
suppose I claim all of mathematics is about first order structures in the sense of model theory, what do you reply
I do not think and model theorist would claim that. The first order model theory has a special place just because there are compactness theorems which give more power to model theory. Shellah’s work has isolated some classes of models of higher order and infinitary logics like AECs where some analogues to the first order situation still hold.
The intersection of model theory and algebraic geometry is more-or-less orthogonal to scheme theory.
Not completely. If one uses noncommutative approach, considering the schemes as their abelian (possibly monoidal) categories of quasicoherent sheaves then one can consider much of their geometry studying definable subcategores and similar objects where model theory is of much use; see for example works of Prest.
I have been interested in categorification of model theory in a similar vain. If developed one could look for appropriate Tannaka duality theorems where both the symmetry object and its representation category would be treated by model theory in the same footing.
The methods used are very different, e.g. o-minimal structures
Those capture it seems the tame topology conjectured by Grothendieck which is much closer to the effective methods and effective intuition of topology than the study of usual topological spaces. This is quite an achievement from the point of view of practical mathematics.
But anyway, model theory is like the study of integrable systems and the study of TQFTs. Everybody know that not all systems are integrable, and everybody knows that most theories are not as simple, symmetric, topological, solvable etc. as most interesting TQFTs, CFTs. But those are very important examples, and over there, in a regular setup, one learns much about the general case. One could say the same of topos theory. Most interesting categories from practical mathematics, even geometry, are not Grothendieck topoi (not even if we allow categorified, elementary and slightly weakened variants). But this is the world where all possible constructions of moduli spaces etc. work, and in other cases, it will be guiding examples.
Re #6: On the other hand, ZFC is a first-order theory…
16: this just says that the “classification” of the models for the set theoretic universe can be studied from the point of view of the classical model theory. Am I right ?
ZFC as we understand it today is a first-order theory, but Zermelo seemed to disagree!
Zermelo at first considered the Skolem paradox a hoax (van Dalen and Ebbinghaus, 2000, p. 148 ff.), and spoke against it starting in 1929. Skolem’s result applies only to what is now called first-order logic, but Zermelo argued against the finitary metamathematics that underlie first-order logic (Kanamori 2004, p. 519 ff.). Zermelo argued that his axioms should instead be studied in second-order logic, a setting in which Skolem’s result does not apply. Zermelo published a second-order axiomatization in 1930 and proved several categoricity results in that context.
See this article, for instance.
What I mean is that the axioms of ZFC (in its modern meaning) are formulated in first-order logic. As a consequence, its model theory can be studied in any Heyting category.
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