Description: Axiom of Replacement. An
axiom scheme of Zermelo-Fraenkel set theory.
Axiom 5 of [TakeutiZaring] p. 19.
It tells us that the image of any set
under a function is also a set (see the variant funimaex 6472). Although
𝜑 may be any wff whatsoever, this
axiom is useful (i.e. its
antecedent is satisfied) when we are given some function and 𝜑
encodes the predicate "the value of the function at 𝑤 is
𝑧".
Thus, 𝜑 will ordinarily have free variables
𝑤
and 𝑧- think
of it informally as 𝜑(𝑤, 𝑧). We prefix 𝜑 with the
quantifier ∀𝑦 in order to "protect" the
axiom from any 𝜑
containing 𝑦, thus allowing us to eliminate any
restrictions on
𝜑. Another common variant is derived
as axrep5 5190, where you can
find some further remarks. A slightly more compact version is shown as
axrep2 5187. A quite different variant is zfrep6 7733, which if used in
place of ax-rep 5184 would also require that the Separation Scheme
axsep 5196
be stated as a separate axiom.
There is a very strong generalization of Replacement that doesn't demand
function-like behavior of 𝜑. Two versions of this
generalization
are called the Collection Principle cp 9512 and the Boundedness Axiom
bnd 9513.
Many developments of set theory distinguish the uses of Replacement from
uses of the weaker axioms of Separation axsep 5196, Null Set axnul 5203, and
Pairing axpr 5326, all of which we derive from Replacement. In
order to
make it easier to identify the uses of those redundant axioms, we
restate them as Axioms ax-sep 5197, ax-nul 5204, and ax-pr 5327 below the
theorems that prove them. (Contributed by NM,
23-Dec-1993.) |