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 8479 and inf2 8480). More
standard versions, which essentially state that there exists a set
containing all the natural numbers, are shown as zfinf2 8499 and omex 8500 and
are based on the (nontrivial) proof of inf3 8492.
This version has the
advantage that when expanded to primitives, it has fewer symbols than
the standard version ax-inf2 8498. Theorem inf0 8478
shows the reverse
derivation of our axiom from a standard one. Theorem inf5 8502
shows a
very short way to state this axiom.
The standard version of Infinity ax-inf2 8498 requires this axiom along
with Regularity ax-reg 8457 for its derivation (as theorem axinf2 8497 below).
In order to more easily identify the normal uses of Regularity, we will
usually reference ax-inf2 8498 instead of this one. The derivation of this
axiom from ax-inf2 8498 is shown by theorem axinf 8501.
Proofs should normally use the standard version ax-inf2 8498 instead of
this axiom. (New usage is discouraged.) (Contributed by NM,
16-Aug-1993.) |