language, signature theory, first-order theory model interpretation structure in model theory elementary embedding type in model theory compactness theorem diagram of a first-order structure definable set (combinatorial) pregeometry ultraproduct, ultraroot, ultrapower indiscernible sequence? Morleyization Morley sequence? Ramsey theorem? Erdos-Rado theorem? Ehrenfeucht-Fraïssé games (back-and-forth games) Fraïssé limits monster model Hrushovski construction? product of theories definable group definable groupoid definable category elimination of imaginaries elimination of hyperimaginaries elimination of quantifiers omega-categorical structure existentially closed model model completeness stable theories geometric stability theory generic predicate? ultracategory ACF DLO countable random graph ACVF? RCF? nonstandard analysis Peano arithmetic Robinson arithmetic Nelson arithmetic Ax-Kochen-Ershov theorem? Keisler-Shelah isomorphism theorem conceptual completeness Makkai duality Los ultraproduct theorem Morley categoricity theorem? omitting types theorem Deligne completeness theorem In the same way that in nonstandard analysis one passes to elementary extensions of the field which realizes more types (in particular the types of infinite and infinitesimal numbers), one can extend the standard model of first-order Peano arithmetic to proper elementary extensions that have infinite natural numbers. These “points at infinity” that live in the nonstandard part tend to embody the uniform behavior of the numbers from the standard part. (For example, by compactness, the twin prime conjecture is true if and only if in some nonstandard model of arithmetic there exists just a single pair of nonstandard twin primes, and similarly Dirichlet’s theorem on arithmetic progressions can be reformulated as saying that for every coprime positive pair of numbers and there exists just one nonstandard prime congruent to modulo .) A nonstandard model of arithmetic is a proper elementary...