Editing Compactness theorem (section)

==Applications==

The compactness theorem has many applications in model theory; a few typical results are sketched here.

===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 />

===Upward Löwenheim–Skolem theorem===

A second application of the compactness theorem shows that any theory that has arbitrarily large finite models, or a single infinite model, has models of arbitrary large [[cardinality]] (this is the [[Upward Löwenheim–Skolem theorem]]). So for instance, there are nonstandard models of [[Peano arithmetic]] with uncountably many 'natural numbers'.  To achieve this, let <math>T</math> be the initial theory and let <math>\kappa</math> be any [[cardinal number]]. Add to the language of <math>T</math> one constant symbol for every element of <math>\kappa.</math> Then add to <math>T</math> a collection of sentences that say that the objects denoted by any two distinct constant symbols from the new collection are distinct (this is a collection of <math>\kappa^2</math> sentences). Since every {{em|finite}} subset of this new theory is satisfiable by a sufficiently large finite model of <math>T,</math> or by any infinite model, the entire extended theory is satisfiable. But any model of the extended theory has cardinality at least <math>\kappa</math>.

===Non-standard analysis===

A third application of the compactness theorem is the construction of [[Non-standard analysis|nonstandard models]] of the real numbers, that is, consistent extensions of the theory of the real numbers that contain "infinitesimal" numbers.  To see this, let <math>\Sigma</math> be a first-order axiomatization of the theory of the real numbers.  Consider the theory obtained by adding a new constant symbol <math>\varepsilon</math> to the language and adjoining to <math>\Sigma</math> the axiom <math>\varepsilon > 0</math> and the axioms <math>\varepsilon < \tfrac{1}{n}</math> for all positive integers <math>n.</math>  Clearly, the standard real numbers <math>\R</math> are a model for every finite subset of these axioms, because the real numbers satisfy everything in <math>\Sigma</math> and, by suitable choice of <math>\varepsilon,</math> can be made to satisfy any finite subset of the axioms about <math>\varepsilon.</math>  By the compactness theorem, there is a model <math>{}^* \R</math> that satisfies <math>\Sigma</math> and also contains an infinitesimal element <math>\varepsilon.</math>  

A similar argument, this time adjoining the axioms <math>\omega > 0, \; \omega > 1, \ldots,</math> etc., shows that the existence of numbers with infinitely large magnitudes cannot be ruled out by any axiomatization <math>\Sigma</math> of the reals.{{sfn|Goldblatt|1998|pages=[https://archive.org/details/lecturesonhyperr00gold_574/page/n12 10]–11}} 

It can be shown that the [[hyperreal number]]s <math>{}^* \R</math> satisfy the [[transfer principle]]:{{sfn|Goldblatt|1998|p=11}} a first-order sentence is true of <math>\R</math> if and only if it is true of <math>{}^* \R.</math>