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Set theory and model theory have many applications outside of logic, in particular in algebra, topology, analysis, ...

On the other hand model theory, in particular after Hrushovski, found many applications in algebraic geometry and Diophantine geometry.

(A) I wonder to know if there are any nontrivial applications of set theory in branches like algebraic geometry, Diophantine geometry, K-theory or number theory (algebraic or analytic)? In particular:

1) are there statements in these fields which are independent from $ZFC$?

2) Are there $ZFC$ provable statements in these fields whose proofs are known just using set theoretic methods?

(B) On the other hand are there any results in set theory whose proofs are based on some techniques from the above quoted fields?

Giving references is appreciated.

(C) Are there any connections between model theory and algebraic or analytic number theory?

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2 Answers 2

I suppose this counts as algebraic geometry, so it would be an example of (A) 1).

Let $R$ be a ring and $D(R)$ its unbounded derived category. Let $D^c(R)$ be the full subcategory of compact objects (in the explicit example below it is spanned by bounded complexes of f.g. projective modules). We say that $D(R)$ satisfies Adams representability if any cohomological functor $D^c(R)^{op}\rightarrow Ab$, i.e. additive and taking exact triangles to exact sequences, is isomorphic to the restriction of a representable functor in $D(R)$ (in particular it extends to the whole $D(R)$), and any natural transformation between restrictions of representable functors $D^c(R)^{op}\rightarrow Ab$ is induced by a morphism in $D(R)$ between the representatives.

Let $\mathbb C\langle x,y\rangle$ be the ring of noncommutative polinomials on two variables. The statement '$D(\mathbb C\langle x,y\rangle)$ satisfies Adams representability' is equivalent to the continuum hypothesis.

You can make similar statements with commutative $R$, they are related to $|\mathbb C|=\aleph_n$ for $n>1$ (still independent of ZFC), this is why I prefered the previous explicit example.

All this follows from :

Failure of Brown representability in derived categories
J. Daniel Christensen, Bernhard Keller, Amnon Neeman
Topology 40 (2001) 1339}1361

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do you know why it's called adams representability? if we stick with commutative spaces (ie commutative rings, schemes, algebraic spaces, stacks or what have you) are there some necessary/sufficient conditions related to the adams condition? –  bananastack May 11 '14 at 4:46
@user125763 Some people call it Brown representability, but this name is usually kept for a weaker property (which is satisfied by all derived categories, BTW). I like to call it Adams representability because, for the stable homotopy category, it was first proved in Adams, J. F. A variant of E. H. Brown's representability theorem. Topology 10 1971 185–198. –  Fernando Muro May 12 '14 at 11:11
It may be worth remarking that this theorem is independent of ZFC but not stronger than ZFC since ZFC can interpret it as a theorem on constructible sets. –  Colin McLarty May 18 '14 at 14:08

(C) Recently applied model theorists have touched many areas of algebra, algebraic geometry, number theory and even analysis structures.

(1) Exponential fields:

Schanuel's conjecture is a conjecture made by Stephen Schanuel in the 1960s:

Given any $n$ complex numbers $z_1,\dots,z_n$ which are linearly independent over the rational numbers $\mathbb{Q}$, the extension field $\mathbb{Q}(z_1,\dots,z_n, \exp(z_1),\dots,\exp(z_n))$ has transcendence degree of at least $n$ over $\mathbb{Q}$.

In 2004, Boris Zilber systematically constructs exponential fields $K_{\exp}$ that are algebraically closed and of characteristic zero, and such that one of these fields exists for each uncountable cardinal. Zilber axiomatises these fields and by using the Hrushovski's construction and techniques inspired by work of Shelah on categoricity in infinitary logics, proves that this theory of "pseudo-exponentiation" has a unique model in each uncountable cardinal. See here and here for more.

(2) Polynomial dynamics:

The connection between algebraic dynamics and the model theory of difference fields was first noticed by Chatzidakis and Hrushovski. A series of three papers entitled "Difference fields and descent in algebraic dynamics". It seems that the first-order theories of algebraically closed difference fields where the automorphism is "generic" are quite nice. See here for more result by Scanlon and Alice Medvedev.

(3) Diophantine geometry:

Hrushovski, Scanlon and their students have worked on model theory and its application in Diophantine geometry. See here for information about applications of model theory in Diophantine geometry.

(4) Algebraic geometry:

The Mordell-Lang conjecture for function fields: Let $k_0\subset K$ be two distinct algebraically closed fields. Let $A$ be an abelian variety defined over $K$, let $X$ be an infinite subvariety of $A$ defined over $K$ and let $\Gamma$ be a subgroup of "finite rank" of $A(K)$. Suppose that $X\cap \Gamma$ is Zariski dense in $X$ and that the stabilizer of $X$ in $A$ is finite. Then there is a subabelian variety $B$ of $A$ and there are $S$, an abelian variety defined over $k_0$, $X_0$ a subvariety of $S$ defined over $k_0$, and a bijective morphism $h$ from $B$ onto $S$, such that $X=a_0 + h^{-1}(X_0)$ for some $a_0$ in $A$.

This theorem is proved by Hrushovski in 1996, see here. For more see this book.

(5) Number theory:

For example see the recent works of Jonathan Pila.

(6) Analysis:

Traditionally model theory is consistent with algebra. But recently, model theorists have been interested in continuous structures that appears in analysis, for example Banach spaces. For more see here.

Model theory has many other application in other fields of mathematics, such as geometric group theory, differential algebra, Berkovich spaces (see recent works of Hrushovski, Loeser, Poonen here and here), approximate groups, etc. (for more see here, here, here and here )

Note: Model theorists have many important and interesting problems in their fields and I believe that the goal of model theory is not necessary to solve the problems of the other fields!

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Dear Mostafa, I do not understand your (1): what is the relation between Schanuel's conjecture and Zilber's costruction ? Thanks –  Joël Sep 22 '14 at 17:47
Dear Joël, Schanuel's conjecture is part of Zilber's axiomatisation, and so the natural conjecture that the unique model of cardinality continuum is actually isomorphic to the complex exponential field implies Schanuel's conjecture. In fact, Zilber shows that this conjecture holds iff another unproven condition on the complex exponentiation field, which Zilber calls exponential-algebraic closedness, hold. –  Mostafa Mirabi Sep 24 '14 at 5:35
@Joël, Boris Zilber in his paper "EXPONENTIAL SUMS EQUATIONS AND THE SCHANUEL CONJECTURE" says : "A uniform version of the Schanuel conjecture is discussed that has some model-theoretical motivation. This conjecture is assumed, and it is proved that any 'non-obviously-contradictory' system of equations in the form of exponential sums with real exponents has a solution." –  Mostafa Mirabi Sep 24 '14 at 5:39
Dear Joel, for more please look at this paper: "Pseudo-exponentiation on algebraically closed fields of characteristic zero". Annals of Pure and Applied Logic, Zilber, Boris (2004). –  Mostafa Mirabi Sep 24 '14 at 5:40

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