replacement, axiom of
- The axiom added to Zermelo's set theory by A. A. Fraenkel (1891–1965), to produce the classical set theory known as ZF. Put in terms of second-order logic, the axiom states that any function whose domain is a set has a range which is also a set. That is, if the arguments of a function form a set, so do the values of the function. This formulation is second-order because it quantifies over functions; in first-order logic the axiom needs to be stated as an axiom schema. See also Zermelo–Fraenkel set theory.
Philosophy dictionary. Academic. 2011.
Look at other dictionaries:
replacement — replacement, axiom of … Philosophy dictionary
Replacement — means: * Axiom schema of replacement * Text replacement, a feature of word processors correcting automatically common misspellings and typos * Replacement rate * Sampling (statistics) with replacementee also*Replacements … Wikipedia
Axiom schema of replacement — In set theory, the axiom schema of replacement is a schema of axioms in Zermelo Fraenkel set theory (ZFC) that asserts that the image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinite… … Wikipedia
Axiom schema of specification — For the separation axioms in topology, see separation axiom. In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of specification, axiom schema of separation, subset axiom scheme or… … Wikipedia
Axiom of empty set — In set theory, the axiom of empty set is one of the axioms of Zermelo–Fraenkel set theory and one of the axioms of Kripke–Platek set theory. Formal statement In the formal language of the Zermelo–Fraenkel axioms, the axiom reads::exist x, forall… … Wikipedia
Axiom of limitation of size — In class theories, the axiom of limitation of size says that for any class C , C is a proper class (a class which is not a set (an element of other classes)) if and only if V (the class of all sets) can be mapped one to one into C .:forall C… … Wikipedia
Axiom schema — In mathematical logic, an axiom schema generalizes the notion of axiom.An axiom schema is a formula in the language of an axiomatic system, in which one or more schematic variables appear. These variables, which are metalinguistic constructs,… … Wikipedia
Axiom of regularity — In mathematics, the axiom of regularity (also known as the axiom of foundation) is one of the axioms of Zermelo Fraenkel set theory and was introduced by harvtxt|von Neumann|1925. In first order logic the axiom reads::forall A (exists B (B in A)… … Wikipedia
Axiom of pairing — In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo Fraenkel set theory. Formal statement In the formal language of the Zermelo Frankel axioms, the … Wikipedia
Constructive set theory — is an approach to mathematical constructivism following the program of axiomatic set theory. That is, it uses the usual first order language of classical set theory, and although of course the logic is constructive, there is no explicit use of… … Wikipedia