Editing First-order logic (section)

===Infinitary logics===
{{Main|Infinitary logic}}
Infinitary logic allows infinitely long sentences. For example, one may allow a conjunction or disjunction of infinitely many formulas, or quantification over infinitely many variables. Infinitely long sentences arise in areas of mathematics including [[topology]] and [[model theory]].

Infinitary logic generalizes first-order logic to allow formulas of infinite length. The most common way in which formulas can become infinite is through infinite conjunctions and disjunctions. However, it is also possible to admit generalized signatures in which function and relation symbols are allowed to have infinite arities, or in which quantifiers can bind infinitely many variables. Because an infinite formula cannot be represented by a finite string, it is necessary to choose some other representation of formulas; the usual representation in this context is a tree. Thus, formulas are, essentially, identified with their parse trees, rather than with the strings being parsed.

The most commonly studied infinitary logics are denoted ''L''<sub>αβ</sub>, where α and β are each either [[cardinal number]]s or the symbol ∞. In this notation, ordinary first-order logic is ''L''<sub>ωω</sub>.
In the logic ''L''<sub>∞ω</sub>, arbitrary conjunctions or disjunctions are allowed when building formulas, and there is an unlimited supply of variables. More generally, the logic that permits conjunctions or disjunctions with less than κ constituents is known as ''L''<sub>κω</sub>. For example, ''L''<sub>ω<sub>1</sub>ω</sub> permits [[countable set|countable]] conjunctions and disjunctions.

The set of free variables in a formula of ''L''<sub>κω</sub> can have any cardinality strictly less than κ, yet only finitely many of them can be in the scope of any quantifier when a formula appears as a subformula of another.<ref>Some authors only admit formulas with finitely many free variables in ''L''<sub>κω</sub>, and more generally only formulas with <&nbsp;λ free variables in ''L''<sub>κλ</sub>.</ref> In other infinitary logics, a subformula may be in the scope of infinitely many quantifiers. For example, in ''L''<sub>κ∞</sub>, a single universal or existential quantifier may bind arbitrarily many variables simultaneously. Similarly, the logic ''L''<sub>κλ</sub> permits simultaneous quantification over fewer than λ variables, as well as conjunctions and disjunctions of size less than κ.