information:classification
Non-standard analysis

A third application of the compactness theorem is the construction of 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 Σ {\displaystyle \Sigma } be a first-order axiomatization of the theory of the real numbers. Consider the theory obtained by adding a new constant symbol ε {\displaystyle \varepsilon } to the language and adjoining to Σ {\displaystyle \Sigma } the axiom ε > 0 {\displaystyle \varepsilon >0} and the axioms ε < 1 n {\displaystyle \varepsilon <{\tfrac {1}{n}}} for all positive integers n. {\displaystyle n.} Clearly, the standard real numbers R {\displaystyle \mathbb {R} } are a model for every finite subset of these axioms, because the real numbers satisfy everything in Σ {\displaystyle \Sigma } and, by suitable choice of ε, {\displaystyle \varepsilon,} can be made to satisfy any finite subset of the axioms about ε. {\displaystyle \varepsilon.} By the compactness theorem, there is a model ∗ R {\displaystyle {}^{*}\mathbb {R} } that satisfies Σ {\displaystyle \Sigma } and also contains an infinitesimal element ε. {\displaystyle \varepsilon.}

A similar argument, this time adjoining the axioms ω > 0, ω > 1, …, {\displaystyle \omega >0,\;\omega >1,\ldots,} etc., shows that the existence of numbers with infinitely large magnitudes cannot be ruled out by any axiomatization Σ {\displaystyle \Sigma } of the reals.

It can be shown that the hyperreal numbers ∗ R {\displaystyle {}^{*}\mathbb {R} } satisfy the transfer principle: a first-order sentence is true of R {\displaystyle \mathbb {R} } if and only if it is true of ∗ R. {\displaystyle {}^{*}\mathbb {R}.}