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 8816 and inf2 8817). More
standard versions, which essentially state that there exists a set
containing all the natural numbers, are shown as zfinf2 8836 and omex 8837 and
are based on the (nontrivial) proof of inf3 8829.
This version has the
advantage that when expanded to primitives, it has fewer symbols than
the standard version ax-inf2 8835. Theorem inf0 8815
shows the reverse
derivation of our axiom from a standard one. Theorem inf5 8839
shows a
very short way to state this axiom.
The standard version of Infinity ax-inf2 8835 requires this axiom along
with Regularity ax-reg 8786 for its derivation (as theorem axinf2 8834 below).
In order to more easily identify the normal uses of Regularity, we will
usually reference ax-inf2 8835 instead of this one. The derivation of this
axiom from ax-inf2 8835 is shown by theorem axinf 8838.
Proofs should normally use the standard version ax-inf2 8835 instead of
this axiom. (New usage is discouraged.) (Contributed by NM,
16-Aug-1993.) |