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An infinitary logic is a logic that allows infinitely long statements and/or infinitely long proofs. The concept was introduced by Zermelo in the 1930s...

arity, i.e. if it has a finite number of input values. Similarly, an infinitary operation is one with an infinite number of input values. In standard...

In mathematics, infinitary combinatorics, or combinatorial set theory, is an extension of ideas in combinatorics to infinite sets. Some of the things studied...

the fact that if a cardinal is weakly compact then a certain related infinitary language satisfies a version of the compactness theorem; see below. The...

the context is clear). It is also possible to generalize the concept to infinitary relations with infinite sequences. When two objects, qualities, classes...

arithmetic combinatorics is the ergodic theory of dynamical systems. Infinitary combinatorics, or combinatorial set theory, is an extension of ideas in...

numbers as its domain. Many extensions of first-order logic, including infinitary logics and higher-order logics, are more expressive in the sense that...

rule. An example of a rule that is not effective in this sense is the infinitary ω-rule. Popular rules of inference in propositional logic include modus...

category Law. Variations include multisorted (or multityped) Lawvere theory, infinitary Lawvere theory, and finite-product theory. Algebraic theory Clone (algebra)...

ternary operation. Generally, the arity is taken to be finite. However, infinitary operations are sometimes considered, in which case the "usual" operations...

sufficient strength. Ordinal analysis allows one to measure precisely the infinitary content of the consistency of theories. For a consistent recursively axiomatized...

structure of sets. Gödel sets can also be used to encode formulas in infinitary languages. Church encoding Description number Gödel numbering for sequences...

a group can be modified by forcing. Hamkins has investigated several infinitary games, including infinite chess, infinite draughts, infinite Hex, and...

contingent truths are largely unknown to humans, Leibniz made appeal to infinitary sufficient reasons, to which God uniquely has access: In contingent truths...

philosophy Region Western philosophy School Analytic philosophy Thesis Infinitary Languages and Descriptive Set Theory (1974) Doctoral advisor Jack Silver...

is the union of an arbitrary collection of sets, sometimes called an infinitary union. If M is a set or class whose elements are sets, then x is an element...

the monoid. A complete monoid is a commutative monoid equipped with an infinitary sum operation Σ I {\displaystyle \Sigma _{I}} for any index set I such...

axiomatisable by a first-order theory. Model theory in higher-order logics or infinitary logics is hampered by the fact that completeness and compactness do not...

extension to infinitary operations. Especially in order theory one finds numerous important variants of distributivity, some of which include infinitary operations...

notation Knuth's up-arrow notation Arrow notation (Ramsey theory), or infinitary combinatorics Arrow notation as a way of representing functions This disambiguation...