Editing Model theory (section)

==Selected applications==

Among the early successes of model theory are Tarski's proofs of quantifier elimination for various algebraically interesting classes, such as the [[real closed field]]s, [[Boolean algebra (structure)|Boolean algebras]] and [[algebraically closed field]]s of a given [[characteristic (algebra)|characteristic]]. Quantifier elimination allowed Tarski to show that the first-order theories of real-closed and algebraically closed fields as well as the first-order theory of Boolean algebras are decidable, classify the Boolean algebras up to elementary equivalence and show that the theories of real-closed fields and algebraically closed fields of a given characteristic are unique. Furthermore, quantifier elimination provided a precise description of definable relations on algebraically closed fields as [[algebraic varieties]] and of the definable relations on real-closed fields as [[semialgebraic sets]]{{sfn|Hodges|1993|pp=68-69}}<ref>{{Cite journal|last1=Doner|first1=John|last2=Hodges|first2=Wilfrid|date=March 1988|title=Alfred Tarski and Decidable Theories|url=http://dx.doi.org/10.2307/2274425|journal=The Journal of Symbolic Logic|volume=53|issue=1|pages=20|doi=10.2307/2274425|jstor=2274425|issn=0022-4812}}</ref>

In the 1960s, the introduction of the [[ultraproduct]] construction led to new applications in algebra. This includes [[James Ax|Ax's]] work on [[pseudofinite field]]s, proving that the theory of finite fields is decidable,<ref>{{Citation|last=Eklof|first=Paul C.|title=Ultraproducts for Algebraists|date=1977|url=http://dx.doi.org/10.1016/s0049-237x(08)71099-1|work=HANDBOOK OF MATHEMATICAL LOGIC|series=Studies in Logic and the Foundations of Mathematics|volume=90|pages=105–137|publisher=Elsevier|doi=10.1016/s0049-237x(08)71099-1|isbn=9780444863881|access-date=2022-01-23}}</ref> and Ax and [[Simon B. Kochen|Kochen]]'s proof of as special case of Artin's conjecture on diophantine equations, the [[Ax–Kochen theorem]].<ref>{{Cite journal|last1=Ax|first1=James|last2=Kochen|first2=Simon|date=1965|title=Diophantine Problems Over Local Fields: I. |journal=American Journal of Mathematics|volume=87|issue=3 |pages=605–630|doi=10.2307/2373065 |jstor=2373065 }}</ref> The ultraproduct construction also led to [[Abraham Robinson]]'s development of [[nonstandard analysis]], which aims to provide a rigorous calculus of [[infinitesimals]].<ref>{{Citation|last1=Cherlin|first1=Greg|title=Ultrafilters and Ultraproducts in Non-Standard Analysis|date=1972|url=http://dx.doi.org/10.1016/s0049-237x(08)71563-5|work=Contributions to Non-Standard Analysis|pages=261–279|publisher=Elsevier|access-date=2022-01-23|last2=Hirschfeld|first2=Joram|series=Studies in Logic and the Foundations of Mathematics|volume=69|doi=10.1016/s0049-237x(08)71563-5|isbn=9780720420654}}</ref>

More recently, the connection between stability and the geometry of definable sets led to several applications from algebraic and diophantine geometry,  including [[Ehud Hrushovski]]'s 1996 proof of the geometric [[Mordell–Lang conjecture]] in all characteristics<ref>Ehud Hrushovski, The  Mordell-Lang  Conjecture for  Function  Fields. [[Journal of the American Mathematical Society]] 9:3 (1996), pp. 667-690.</ref> In 2001, similar methods were used to prove a generalisation of the Manin-Mumford conjecture.
In 2011, [[Jonathan Pila]] applied techniques around [[O-minimal theory|o-minimality]] to prove the  [[André–Oort conjecture]] for products of Modular curves.<ref>{{cite journal|first=Jonathan|last=Pila|title=O-minimality and the André–Oort conjecture for ''C''<sup>''n''</sup>. |journal=[[Annals of Mathematics]] |volume=173|issue=3 |year=2011| pages=1779–1840| doi=10.4007/annals.2011.173.3.11}}</ref>

In a separate strand of inquiries that also grew around stable theories, Laskowski showed in 1992 that  [[NIP (model theory)|NIP theories]] describe exactly those definable classes that are [[Probably approximately correct learning|PAC-learnable]] in machine learning theory. This has led to several interactions between these separate areas. In 2018, the correspondence was extended as Hunter and Chase showed that stable theories correspond to [[Online machine learning|online learnable classes]].<ref>{{Cite journal|last1=CHASE|first1=HUNTER|last2=FREITAG|first2=JAMES|title=Model Theory and Machine Learning|date=2019-02-15|url=http://dx.doi.org/10.1017/bsl.2018.71|journal=The Bulletin of Symbolic Logic|volume=25|issue=3|pages=319–332|doi=10.1017/bsl.2018.71|arxiv=1801.06566|s2cid=119689419|issn=1079-8986}}</ref>