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...

of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation (Aussonderungsaxiom), subset axiom, axiom of...

an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom. An axiom schema is a formula in the metalanguage of an axiomatic...

proposed replacing the axiom schema of specification with the axiom schema of replacement. Appending this schema, as well as the axiom of regularity (first...

the standard formulation of the Zermelo–Fraenkel set theory, the axiom of pairing follows from the axiom schema of replacement applied to any given set...

number of cardinalities. Together with the axiom schema of replacement, the axiom of union implies that one can form the union of a family of sets indexed...

existence of the empty set is a theorem. If separation is not postulated as an axiom schema, but derived as a theorem schema from the schema of replacement (as...

Halmos' axiom (schema) of substitution is equivalent to the axiom schema of replacement, in the presence of the other axioms. Additionally, axioms 1.-8. are...

Axiom of extensionality Axiom of empty set Axiom of pairing Axiom of union Axiom of infinity Axiom schema of replacement Axiom of power set Axiom of regularity...

element of f(n) for each n. Define S = {f(n): n a natural number}, the range of f, which can be seen to be a set from the axiom schema of replacement. Applying...

\ldots ,x_{n})].} Then the axiom schema of replacement is replaced by a single axiom that uses a class. Finally, ZFC's axiom of extensionality is modified...

sets using the axiom schemas of specification and replacement, as well as the axiom of power set, introduces impredicativity, a type of circularity, into...

and related operating systems Axiom schema of replacement, a schema of axioms in Zermelo–Fraenkel set theory Replacement rates, in population fertility...

showed that any proof of the theorem in Zermelo–Fraenkel set theory must make repeated use of instances of the axiom schema of replacement. Later results showed...

as an example the axiom schema of replacement in Zermelo–Fraenkel set theory. (This example uses mathematical symbols.) This schema states (in one form)...

If the axiom schema of replacement is added as another axiom, the resulting theory is equivalent to ZFC. William Lawvere, An elementary theory of the category...

version of this axiom resembles the axiom schema of replacement, and embodies the class function F. The next section explains how Limitation of Size is...

situation has a clearly defined beginning or end Boundedness axiom, the axiom schema of replacement Bounded deformation, a function whose distributional derivatives...

use the Axiom of Infinity combined with the Axiom schema of specification. Let I {\displaystyle I} be an inductive set guaranteed by the Axiom of Infinity...

called the axiom schema of set induction. The principle implies transfinite induction and recursion. It may also be studied in a general context of induction...

member of itself, and circular chains of membership are impossible. Axiom schema of replacement: Let the domain of the class function F {\displaystyle F}...

commonly have Axiom schema of Replacement, sometimes restricted to bounded formulas. However, when other axioms are dropped, this schema is actually often...

axiom is replaced by an axiom schema. The notion of "first order formula" was not known in 1908 when Zermelo published his axiom system, and he later rejected...

from the schema all instances of the Replacement Axiom for j-formulas". Thus, the wholeness axiom differs from Reinhardt cardinals (another way of providing...

} (Axioms 2, 3, and 4 are axiom schemas—families of similar axioms. Instances of Axiom 2 and Axiom 3 vary only by the types of variables and...

axiom implies the axioms of replacement, separation, union, and global choice. It is equivalent to the combination of replacement, union, and global...

conventional Von Neumann definition of ordinal numbers. Let τ(x) be a first-order term. Replacement: An axiom schema. For any collection a, ∀x∈a[τ(x) is...

containing precisely those elements x for which φ(x) holds. (This is an axiom schema.) Axiom of Δ0-collection: Given any Δ0 formula φ(x, y), if for every set x...

identity. S has no axiom of extensionality and identity is absent from the other S axioms. Identity does appear in the axiom schema distinguishing S+ from...

mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty...