Axiom ax-inf 8076

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 8060 and inf2 8061). More standard versions, which essentially state that there exists a set containing all the natural numbers, are shown as zfinf2 8080 and omex 8081 and are based on the (nontrivial) proof of inf3 8073. This version has the advantage that when expanded to primitives, it has fewer symbols than the standard version ax-inf2 8079. Theorem inf0 8059 shows the reverse derivation of our axiom from a standard one. Theorem inf5 8083 shows a very short way to state this axiom.

The standard version of Infinity ax-inf2 8079 requires this axiom along with Regularity ax-reg 8039 for its derivation (as theorem axinf2 8078 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2 8079 instead of this one. The derivation of this axiom from ax-inf2 8079 is shown by theorem axinf 8082.

Proofs should normally use the standard version ax-inf2 8079 instead of this axiom. (New usage is discouraged.) (Contributed by NM, 16-Aug-1993.)

Assertion

Ref	Expression
ax-inf

Distinct variable group:   ,,,

Detailed syntax breakdown of Axiom ax-inf

Step	Hyp	Ref	Expression
1		vx	. . . 4
2		vy	. . . 4
3	1, 2	wel 1819	. . 3
4		vz	. . . . . 6
5	4, 2	wel 1819	. . . . 5
6		vw	. . . . . . . 8
7	4, 6	wel 1819	. . . . . . 7
8	6, 2	wel 1819	. . . . . . 7
9	7, 8	wa 369	. . . . . 6
10	9, 6	wex 1612	. . . . 5
11	5, 10	wi 4	. . . 4
12	11, 4	wal 1393	. . 3
13	3, 12	wa 369	. 2
14	13, 2	wex 1612	1

This axiom is referenced by:  zfinf  8077