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 5330). 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 in order to
"protect" the axiom from any
containing , thus
allowing us to eliminate any restrictions on
. This
makes the axiom usable in a formalization that omits the
logically redundant axiom ax-17 1603. Another common variant is derived
as axrep5 4136, where you can find some further remarks. A
slightly more
compact version is shown as axrep2 4133. A quite different variant is
zfrep6 5748, which if used in place of ax-rep 4131 would also require that
the Separation Scheme axsep 4140 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 cp 7561 and the Boundedness Axiom
bnd 7562.
Many developments of set theory distinguish the uses of Replacement from
uses the weaker axioms of Separation axsep 4140, Null Set axnul 4148, and
Pairing axpr 4213, 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 4141, ax-nul 4149, and ax-pr 4214 below the
theorems that prove them. (Contributed by NM,
23-Dec-1993.) |