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 𝑥, an infinite set
𝑦 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 8212
and
inf2 8213). More standard versions, which essentially
state that there
exists a set containing all the natural numbers, are shown as zfinf2 8232
and omex 8233 and are based on the (nontrivial) proof of inf3 8225.
This
version has the advantage that when expanded to primitives, it has fewer
symbols than the standard version ax-inf2 8231. Theorem inf0 8211
shows the
reverse derivation of our axiom from a standard one. Theorem inf5 8235
shows a very short way to state this axiom.
The standard version of Infinity ax-inf2 8231 requires this axiom along
with Regularity ax-reg 8190 for its derivation (as theorem axinf2 8230
below). In order to more easily identify the normal uses of Regularity,
we will usually reference ax-inf2 8231 instead of this one. The derivation
of this axiom from ax-inf2 8231 is shown by theorem axinf 8234.
Proofs should normally use the standard version ax-inf2 8231 instead of
this axiom. (New usage is discouraged.) (Contributed by NM,
16-Aug-1993.) |