|Description: Axiom of Replacement. An
axiom scheme of Zermelo-Fraenkel set theory.
Axiom 5 of [TakeutiZaring] p. 19.
It tells us that that the image of
any set under a function is also a set (see the variant funimaex 4596).
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 ."
ordinarily have free variables and - think
of it informally as . We prefix
quantifier in order to
"protect" the axiom from any
containing , thus
allowing us to eliminate any restrictions on
makes the axiom usable in a formalization that omits the
logically redundant axiom ax-17 1605. Another common variant is derived
as axrep5 3601, where you can find some further remarks. A
compact version is shown as axrep2 3598. A quite different variant is
zfrep6 4641, which if used in place of ax-rep 3596 would also require that
the Separation Scheme axsep 3605 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 6091 and the Boundedness Axiom
Many developments of set theory distinguish the uses of Replacement from
uses the weaker axioms of Separation axsep 3605, Null Set axnul 3612, and
Pairing axpr 3686, all of which we derive from Replacement. In
make it easier to identify the uses of those redundant axioms, we
restate them as axioms ax-sep 3606, ax-nul 3613, and ax-pr 3687 below the
theorems that prove them.