Axiom ax-inf 4684

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 4669 and inf2 4670). More standard versions, which essentially state that there exists a set containing all the natural numbers, are shown as zfinf 4688 and omex 4689 and are based on the (nontrivial) proof of inf3 4682. Our version has the advantage that when expanded to primitives, it has fewer symbols than the standard version ax-inf2 4687. Theorem inf0 4668 shows the reverse derivation of our axiom from a standard one. Theorem inf5 4690 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 1014 and ax-9 1006) that at least one thing exists.

The standard version of Infinity ax-inf2 4687 requires this axiom along with Regularity ax-reg 4653 for its derivation (as theorem axinf2 4686 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2 4687 instead of this one.

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 996	. . . 4 class x
3		vy	. . . . 5 set y
4	3	cv 996	. . . 4 class y
5	2, 4	wcel 999	. . 3 wff x e. y
6		vz	. . . . . . 7 set z
7	6	cv 996	. . . . . 6 class z
8	7, 4	wcel 999	. . . . 5 wff z e. y
9		vw	. . . . . . . . 9 set w
10	9	cv 996	. . . . . . . 8 class w
11	7, 10	wcel 999	. . . . . . 7 wff z e. w
12	10, 4	wcel 999	. . . . . . 7 wff w e. y
13	11, 12	wa 230	. . . . . 6 wff (z e. w /\ w e. y)
14	13, 9	wex 1021	. . . . 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 995	. . 3 wff A.z(z e. y -> E.w(z e. w /\ w e. y))
17	5, 16	wa 230	. 2 wff (x e. y /\ A.z(z e. y -> E.w(z e. w /\ w e. y)))
18	17, 3	wex 1021	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:  axinf 4685

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