Editing Infinitary logic (section)

==Completeness, compactness, and strong completeness==
A theory is any set of sentences. The truth of statements in models are defined by recursion and will agree with the definition for finitary logic where both are defined. Given a theory ''T'' a sentence is said to be valid for the theory ''T'' if it is true in all models of ''T''.

A logic in the language <math>L_{\alpha , \beta}</math> is complete if for every sentence ''S'' valid in every model there exists a proof of ''S''. It is strongly complete if for any theory ''T'' for every sentence ''S'' valid in ''T'' there is a proof of ''S'' from ''T''. An infinitary logic can be complete without being strongly complete.

A cardinal <math>\kappa \neq \omega</math> is [[weakly compact cardinal|weakly compact]] when for every theory ''T'' in <math>L_{\kappa , \kappa}</math> containing at most <math>\kappa</math> many formulas, if every ''S'' <math>\subseteq</math> ''T'' of cardinality less than <math>\kappa</math> has a model, then ''T'' has a model. A cardinal <math>\kappa \neq \omega</math> is [[strongly compact cardinal|strongly compact]] when for every theory ''T'' in <math>L_{\kappa , \kappa}</math>, without restriction on size, if every ''S'' <math>\subseteq</math> ''T'' of cardinality less than <math>\kappa</math> has a model, then ''T'' has a model.