**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 8127
and
inf2 8128). More standard versions, which essentially
state that there
exists a set containing all the natural numbers, are shown as zfinf2 8147
and omex 8148 and are based on the (nontrivial) proof of inf3 8140.
This
version has the advantage that when expanded to primitives, it has fewer
symbols than the standard version ax-inf2 8146. Theorem inf0 8126
shows the
reverse derivation of our axiom from a standard one. Theorem inf5 8150
shows a very short way to state this axiom.
The standard version of Infinity ax-inf2 8146 requires this axiom along
with Regularity ax-reg 8107 for its derivation (as theorem axinf2 8145
below). In order to more easily identify the normal uses of Regularity,
we will usually reference ax-inf2 8146 instead of this one. The derivation
of this axiom from ax-inf2 8146 is shown by theorem axinf 8149.
Proofs should normally use the standard version ax-inf2 8146 instead of
this axiom. (New usage is discouraged.) (Contributed by NM,
16-Aug-1993.) |