axiom schema of replacement

The axiom schemata of replacement and separation each contain infinitely many instances.

Let us take as an example the axiom schema of replacement in Zermelo-Fraenkel set theory.

Axiom schema of replacement: Let the domain of the function be the set .

Includes no equivalents of Choice or the axiom schema of Replacement.

Strong collection schema: This is the constructive replacement for the axiom schema of replacement.

The axiom schema of replacement is not necessary for the proofs of most theorems of ordinary mathematics.

The axiom schema of separation can almost be derived from the axiom schema of replacement.

The axiom schema of collection is closely related to and frequently confused with the axiom schema of replacement.

Axiom schema of replacement that is part of the standard ZFC axiomatization of set theory.

They also independently proposed replacing the axiom schema of specification with the axiom schema of replacement.

This version of the axiom schema of replacement is now suitable for use in a formal language that doesn't allow the introduction of new function symbols.

In some other axiomatizations of ZF, this axiom is redundant in that it follows from the axiom schema of replacement.

The axiom schema of collection is equivalent to the axiom schema of replacement over the remainder of the ZF axioms.

The import of V is that of the axiom schema of replacement in NBG and ZFC.

Boolos then altered Spec to obtain a variant of S he called S+, such that the axiom schema of replacement is derivable in S+ + Extensionality.

Although the axiom schema of replacement is a standard axiom in set theory today, it is often omitted from systems of type theory and foundation systems in topos theory.

The axiom schema of replacement states that if F is a definable class function, as above, and A is any set, then the image F[A] is also a set.

This axiom is part of Z, but is redundant in ZF because it follows from the axiom schema of replacement, if we are given a set with at least two elements.

Much of the power of ZFC, including the axiom of regularity and the axiom schema of replacement, is included primarily to facilitate the study of the set theory itself.

It implies the axiom schema of specification, axiom schema of replacement, axiom of global choice, and even, as noticed later by Azriel Levy, axiom of union at one stroke.

Nevertheless, in 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 with two or more elements, and thus it is sometimes omitted.

The formal 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 stronger than the usual forms of the axiom of choice.

Because at least one such infinite schema is required (ZFC is not finitely axiomatizable), this shows that the axiom schema of replacement can stand as the only infinite axiom schema in ZFC if desired.

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.

In its place, we find the usual axiom of choice for sets, and the following form of the axiom schema of replacement: if the class F is a function whose domain is a set, the range of F is also a set .