Editing Compactness theorem (section)

===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>.