Editing Mathematical logic (section)

=== Other classical logics ===
Many logics besides first-order logic are studied.  These include [[infinitary logics]], which allow for formulas to provide an infinite amount of information, and [[higher-order logic]]s, which include a portion of set theory directly in their semantics.

The most well studied infinitary logic is <math>L_{\omega_1,\omega}</math>. In this logic, quantifiers may only be nested to finite depths, as in first-order logic, but formulas may have finite or countably infinite conjunctions and disjunctions within them. Thus, for example, it is possible to say that an object is a whole number using a formula of <math>L_{\omega_1,\omega}</math> such as
:<math>(x = 0) \lor (x = 1) \lor (x = 2) \lor \cdots.</math>

Higher-order logics allow for quantification not only of elements of the [[domain of discourse]], but subsets of the domain of discourse, sets of such subsets, and other objects of higher type. The semantics are defined so that, rather than having a separate domain for each higher-type quantifier to range over, the quantifiers instead range over all objects of the appropriate type.  The logics studied before the development of first-order logic, for example Frege's logic, had similar set-theoretic aspects. Although higher-order logics are more expressive, allowing complete axiomatizations of structures such as the natural numbers, they do not satisfy analogues of the completeness and compactness theorems from first-order logic, and are thus less amenable to proof-theoretic analysis.

Another type of logics are '''{{vanchor|fixed-point logic}}s''' that allow [[inductive definition]]s, like one writes for [[primitive recursive function]]s.

One can formally define an extension of first-order logic &mdash; a notion which encompasses all logics in this section because they behave like first-order logic in certain fundamental ways, but does not encompass all logics in general, e.g. it does not encompass intuitionistic, modal or [[fuzzy logic]].

[[Lindström's theorem]] implies that the only extension of first-order logic satisfying both the [[compactness theorem]] and the [[Löwenheim–Skolem theorem#Downward part|downward Löwenheim–Skolem theorem]] is first-order logic.