Editing Infinitary logic (section)

==Complete infinitary logics==
Two infinitary logics stand out in their completeness. These are the logics of <math>L_{\omega , \omega}</math> and <math>L_{\omega_1 , \omega}</math>. The former is standard finitary first-order logic and the latter is an infinitary logic that only allows statements of countable size.

The logic of <math>L_{\omega , \omega}</math> is also strongly complete, compact and strongly compact.

The logic of <math>L_{\omega_1, \omega}</math> fails to be compact, but it is complete (under the axioms given above). Moreover, it satisfies a variant of the [[Craig interpolation]] property.

If the logic of <math>L_{\alpha, \alpha}</math> is strongly complete (under the axioms given above) then <math>\alpha</math> is strongly compact (because proofs in these logics cannot use <math>\alpha</math> or more of the given axioms).