In mathematical logic, an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom.
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👉 Axiom schema in the context of Morse–Kelley set theory
In the foundations of mathematics, Morse–Kelley set theory (MK), Kelley–Morse set theory (KM), Morse–Tarski set theory (MT), Quine–Morse set theory (QM) or the system of Quine and Morse is a first-order axiomatic set theory that is closely related to von Neumann–Bernays–Gödel set theory (NBG). While von Neumann–Bernays–Gödel set theory restricts the bound variables in the schematic formula appearing in the axiom schema of Class Comprehension to range over sets alone, Morse–Kelley set theory allows these bound variables to range over proper classes as well as sets, as first suggested by Quine in 1940 for his system ML.
Morse–Kelley set theory is named after mathematicians John L. Kelley and Anthony Morse and was first set out by Wang (1949) and later in an appendix to Kelley's textbook General Topology (1955), a graduate level introduction to topology. Kelley said the system in his book was a variant of the systems due to Thoralf Skolem and Morse. Morse's own version appeared later in his book A Theory of Sets (1965).
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Axiom schema in the context of Unrestricted comprehension
In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation (Aussonderungsaxiom), subset axiom, axiom of class construction, or axiom schema of restricted comprehension is an axiom schema. Essentially, it says that any definable subclass of a set is a set.
Some mathematicians call it the axiom schema of comprehension, although others use that term for unrestricted comprehension, discussed below.
Axiom schema in the context of New Foundations
In mathematical logic, New Foundations (NF) is a non-well-founded, finitely axiomatizable set theory conceived by Willard Van Orman Quine as a simplification of the theory of types of Principia Mathematica.