**Description: **Axiom of Infinity. An
axiom of Zermelo-Fraenkel set theory. This axiom
is the gateway to "Cantor's paradise" (an expression coined by
Hilbert). It asserts that given a starting set , an infinite set
built from it
exists. Although our version is apparently not
given in the literature, it is similar to, but slightly shorter than,
the Axiom of Infinity in [FreydScedrov] p. 283 (see inf1 7566
and
inf2 7567). More standard versions, which essentially
state that there
exists a set containing all the natural numbers, are shown as zfinf2 7586
and omex 7587 and are based on the (nontrivial) proof of inf3 7579.
This
version has the advantage that when expanded to primitives, it has fewer
symbols than the standard version ax-inf2 7585. Theorem inf0 7565
shows the
reverse derivation of our axiom from a standard one. Theorem inf5 7589
shows a very short way to state this axiom.
The standard version of Infinity ax-inf2 7585 requires this axiom along
with Regularity ax-reg 7549 for its derivation (as theorem axinf2 7584
below). In order to more easily identify the normal uses of Regularity,
we will usually reference ax-inf2 7585 instead of this one. The derivation
of this axiom from ax-inf2 7585 is shown by theorem axinf 7588.
Proofs should normally use the standard version ax-inf2 7585 instead of
this axiom. (New usage is discouraged.) (Contributed by NM,
16-Aug-1993.) |