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 funimaex5671). Although
may be any wff whatsoever, this axiom is
useful (i.e. its
antecedent is satisfied) when we are given some function and
encodes the predicate "the value of the function at is ."
Thus, will ordinarily have free variables
and - think
of it informally as (,). We prefix with the
quantifier A. in order to "protect" the axiom from
any
containing , thus allowing us to
eliminate any restrictions on
. Another common variant is derived as axrep54568, where you can
find some further remarks. A slightly more compact version is shown as
axrep24565. A quite different variant is zfrep66768, which if used in
place of ax-rep4563 would also require that the Separation Scheme
axsep4572
be stated as a separate axiom.
There is very a strong generalization of Replacement that doesn't demand
function-like behavior of . Two versions of this generalization
are called the Collection Principle cp8330 and the Boundedness Axiom
bnd8331.
Many developments of set theory distinguish the uses of Replacement from
uses the weaker axioms of Separation axsep4572, Null Set axnul4580, and
Pairing axpr4690, 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-sep4573, ax-nul4581, and ax-pr4691 below the
theorems that prove them. (Contributed by NM,
23-Dec-1993.)