Editing Compactness theorem (section)

===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>