Axiom ax-inf 5728

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 5713 and inf2 5714). More standard versions, which essentially state that there exists a set containing all the natural numbers, are shown as zfinf2 5732 and omex 5733 and are based on the (nontrivial) proof of inf3 5726. Our version has the advantage that when expanded to primitives, it has fewer symbols than the standard version ax-inf2 5731. Theorem inf0 5712 shows the reverse derivation of our axiom from a standard one. Theorem inf5 5735 shows a very short way to state this axiom.

An interesting property of our version is that, unlike the standard version, it does not assert the independent existence of any set; it needs a starting set x. Since none of our other ZFC axioms assert the independent existence of any set, we would need to add an axiom of existence such as Axiom 0 of [Kunen] p. 10 if we were to use a "free logic" predicate calculus that (unlike ours) does not assert (as we do with ax-4 1319 and ax-9 1307) that at least one thing exists.

The standard version of Infinity ax-inf2 5731 requires this axiom along with Regularity ax-reg 5695 for its derivation (as theorem axinf2 5730 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2 5731 instead of this one. The derivation of this axiom from ax-inf2 5731 is shown by theorem axinf 5734.

Assertion

Ref	Expression
ax-inf	|- E.y(x e. y /\ A.z(z e. y -> E.w(z e. w /\ w e. y)))

Distinct variable group:   x,y,z,w

Detailed syntax breakdown of Axiom ax-inf

Step	Hyp	Ref	Expression
1		vx	. . . . 5 set x
2	1	cv 1297	. . . 4 class x
3		vy	. . . . 5 set y
4	3	cv 1297	. . . 4 class y
5	2, 4	wcel 1300	. . 3 wff x e. y
6		vz	. . . . . . 7 set z
7	6	cv 1297	. . . . . 6 class z
8	7, 4	wcel 1300	. . . . 5 wff z e. y
9		vw	. . . . . . . . 9 set w
10	9	cv 1297	. . . . . . . 8 class w
11	7, 10	wcel 1300	. . . . . . 7 wff z e. w
12	10, 4	wcel 1300	. . . . . . 7 wff w e. y
13	11, 12	wa 240	. . . . . 6 wff (z e. w /\ w e. y)
14	13, 9	wex 1326	. . . . 5 wff E.w(z e. w /\ w e. y)
15	8, 14	wi 3	. . . 4 wff (z e. y -> E.w(z e. w /\ w e. y))
16	15, 6	wal 1296	. . 3 wff A.z(z e. y -> E.w(z e. w /\ w e. y))
17	5, 16	wa 240	. 2 wff (x e. y /\ A.z(z e. y -> E.w(z e. w /\ w e. y)))
18	17, 3	wex 1326	1 wff E.y(x e. y /\ A.z(z e. y -> E.w(z e. w /\ w e. y)))

This axiom is referenced by:  zfinf 5729

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