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In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZF) 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 sets in ZF.
The axiom schema is motivated by the idea that whether a class is a set depends only on the cardinality of the class, not on the rank of its elements. Thus, if one class is "small enough" to be a set, and there is a surjection from that class to a second class, the axiom states that the second class is also a set. However, because ZFC only speaks of sets, not proper classes, the schema is stated only for definable surjections, which are identified with their defining formulas.
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of axiomatic set theory, the axiomschemaof specification, also known as the axiomschemaof separation (Aussonderung Axiom), subset axiom or axiom schema...
an axiomschema (plural: axiom schemata or axiomschemas) generalizes the notion ofaxiom. An axiomschema is a formula in the metalanguage of an axiomatic...
Axiomof extensionality Axiomof empty set Axiomof pairing Axiomof union Axiomof infinity AxiomschemaofreplacementAxiomof power set Axiomof regularity...
existence of the empty set is a theorem. If separation is not postulated as an axiomschema, but derived as a theorem schema from the schemaofreplacement (as...
number of cardinalities. Together with the axiomschemaofreplacement, the axiomof union implies that one can form the union of a family of sets indexed...
sets using the axiomschemasof specification and replacement, as well as the axiomof power set, introduces impredicativity, a type of circularity, into...
commonly have AxiomschemaofReplacement, sometimes restricted to bounded formulas. However, when other axioms are dropped, this schema is actually often...
use the Axiomof Infinity combined with the Axiomschemaof specification. Let I {\displaystyle I} be an inductive set guaranteed by the Axiomof Infinity...
as an example the axiomschemaofreplacement in Zermelo–Fraenkel set theory. (This example uses mathematical symbols.) This schema states (in one form)...
the axiomof choice, abbreviated AC or AoC, is an axiomof set theory equivalent to the statement that a Cartesian product of a collection of non-empty...
situation has a clearly defined beginning or end Boundedness axiom, the axiomschemaofreplacement Bounded deformation, a function whose distributional derivatives...
subclass of the preceding set, so it is a set by the axiomschemaof separation. The class of all order types of well-orderings in W is a set by the axiom schema...
In mathematics, the axiomof determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962...
axiom is replaced by an axiomschema. The notion of "first order formula" was not known in 1908 when Zermelo published his axiom system, and he later rejected...
"stand-ins" for strings; this form of notation is called an "axiomschema" (i.e., there is a countable number of specific forms the notation could take)...
schema of unrestricted comprehension is weakened to the axiomschemaof specification or axiomschemaof separation, If P is a property, then for any set X...
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