Editing Compactness theorem (section)

===Robinson's principle===

The compactness theorem implies the following result, stated by [[Abraham Robinson]] in his 1949 dissertation. 

[[Robinson's principle]]:{{sfn|Marker|2002|pp=40-43}}{{sfn|Gowers|Barrow-Green|Leader|2008|pp=639-643}} If a first-order sentence holds in every [[Field (mathematics)|field]] of [[Characteristic (algebra)|characteristic]] zero, then there exists a constant <math>p</math> such that the sentence holds for every field of characteristic larger than <math>p.</math> This can be seen as follows: suppose <math>\varphi</math> is a sentence that holds in every field of characteristic zero. Then its negation <math>\lnot \varphi,</math> together with the field axioms and the infinite sequence of sentences 
<math display=block>1 + 1 \neq 0, \;\; 1 + 1 + 1 \neq 0, \; \ldots</math>
is not [[Satisfiability|satisfiable]] (because there is no field of characteristic 0 in which <math>\lnot \varphi</math> holds, and the infinite sequence of sentences ensures any model would be a field of characteristic 0). Therefore, there is a finite subset <math>A</math> of these sentences that is not satisfiable. <math>A</math> must contain <math>\lnot \varphi</math> because otherwise it would be satisfiable. Because adding more sentences to <math>A</math> does not change unsatisfiability, we can assume that <math>A</math> contains the field axioms and, for some <math>k,</math> the first <math>k</math> sentences of the form <math>1 + 1 + \cdots + 1 \neq 0.</math> Let <math>B</math> contain all the sentences of <math>A</math> except <math>\lnot \varphi.</math> Then any field with a characteristic greater than <math>k</math> is a model of <math>B,</math> and <math>\lnot \varphi</math> together with <math>B</math> is not satisfiable. This means that <math>\varphi</math> must hold in every model of <math>B,</math> which means precisely that <math>\varphi</math> holds in every field of characteristic greater than <math>k.</math> This completes the proof. 

The [[Lefschetz principle]], one of the first examples of a [[transfer principle]], extends this result. A first-order sentence <math>\varphi</math> in the language of [[Ring (mathematics)|rings]] is true in {{em|some}} (or equivalently, in {{em|every}}) [[algebraically closed]] field of characteristic 0 (such as the [[complex number]]s for instance) if and only if there exist infinitely many primes <math>p</math> for which <math>\varphi</math> is true in {{em|some}} algebraically closed field of characteristic <math>p,</math> in which case <math>\varphi</math> is true in {{em|all}} algebraically closed fields of sufficiently large non-0 characteristic <math>p.</math>{{sfn|Marker|2002|pp=40-43}} 
One consequence is the following special case of the [[Ax–Grothendieck theorem]]: all [[injective map|injective]] [[Complex number|complex]] [[polynomial]]s <math>\Complex^n \to \Complex^n</math> are [[Surjective map|surjective]]{{sfn|Marker|2002|pp=40-43}} (indeed, it can even be shown that its inverse will also be a polynomial).<ref name=Tao2009AxGrothendieck>{{cite web|last=Terence|first=Tao|title=Infinite fields, finite fields, and the Ax-Grothendieck theorem|date=7 March 2009|url=https://terrytao.wordpress.com/2009/03/07/infinite-fields-finite-fields-and-the-ax-grothendieck-theorem/}}</ref> In fact, the surjectivity conclusion remains true for any injective polynomial <math>F^n \to F^n</math> where <math>F</math> is a finite field or the algebraic closure of such a field.<ref name=Tao2009AxGrothendieck />