**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 8145
and
inf2 8146). More standard versions, which essentially
state that there
exists a set containing all the natural numbers, are shown as zfinf2 8165
and omex 8166 and are based on the (nontrivial) proof of inf3 8158.
This
version has the advantage that when expanded to primitives, it has fewer
symbols than the standard version ax-inf2 8164. Theorem inf0 8144
shows the
reverse derivation of our axiom from a standard one. Theorem inf5 8168
shows a very short way to state this axiom.
The standard version of Infinity ax-inf2 8164 requires this axiom along
with Regularity ax-reg 8125 for its derivation (as theorem axinf2 8163
below). In order to more easily identify the normal uses of Regularity,
we will usually reference ax-inf2 8164 instead of this one. The derivation
of this axiom from ax-inf2 8164 is shown by theorem axinf 8167.
Proofs should normally use the standard version ax-inf2 8164 instead of
this axiom. (New usage is discouraged.) (Contributed by NM,
16-Aug-1993.) |