Axiom ax-inf 4594

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 4579 and inf2 4580). More standard versions, which essentially state that there exists a set containing all the natural numbers, are shown as zfinf 4598 and omex 4599 and are based on the (nontrivial) proof of inf3 4592. Our version has the advantage that when expanded to primitives, it has fewer symbols than the standard version ax-inf2 4597. Theorem inf0 4578 shows the reverse derivation of our axiom from a standard one. Theorem inf5 4600 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-4 970 and ax-9 962) that at least one thing exists.

The standard version of Infinity ax-inf2 4597 requires this axiom along with Regularity ax-reg 4565 for its derivation (as theorem axinf2 4596 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2 4597 instead of this one.

Assertion

Ref	Expression
ax-inf	⊢ ∃y(x ∈ y ⋀ ∀z(z ∈ y → ∃w(z ∈ w ⋀ w ∈ 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 952	. . . 4 class x
3		vy	. . . . 5 set y
4	3	cv 952	. . . 4 class y
5	2, 4	wcel 955	. . 3 wff x ∈ y
6		vz	. . . . . . 7 set z
7	6	cv 952	. . . . . 6 class z
8	7, 4	wcel 955	. . . . 5 wff z ∈ y
9		vw	. . . . . . . . 9 set w
10	9	cv 952	. . . . . . . 8 class w
11	7, 10	wcel 955	. . . . . . 7 wff z ∈ w
12	10, 4	wcel 955	. . . . . . 7 wff w ∈ y
13	11, 12	wa 223	. . . . . 6 wff (z ∈ w ⋀ w ∈ y)
14	13, 9	wex 977	. . . . 5 wff ∃w(z ∈ w ⋀ w ∈ y)
15	8, 14	wi 3	. . . 4 wff (z ∈ y → ∃w(z ∈ w ⋀ w ∈ y))
16	15, 6	wal 951	. . 3 wff ∀z(z ∈ y → ∃w(z ∈ w ⋀ w ∈ y))
17	5, 16	wa 223	. 2 wff (x ∈ y ⋀ ∀z(z ∈ y → ∃w(z ∈ w ⋀ w ∈ y)))
18	17, 3	wex 977	1 wff ∃y(x ∈ y ⋀ ∀z(z ∈ y → ∃w(z ∈ w ⋀ w ∈ y)))

This axiom is referenced by:  axinf 4595

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