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Axiom ax-inf 8702
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 8686 and inf2 8687). More standard versions, which essentially state that there exists a set containing all the natural numbers, are shown as zfinf2 8706 and omex 8707 and are based on the (nontrivial) proof of inf3 8699. This version has the advantage that when expanded to primitives, it has fewer symbols than the standard version ax-inf2 8705. Theorem inf0 8685 shows the reverse derivation of our axiom from a standard one. Theorem inf5 8709 shows a very short way to state this axiom.

The standard version of Infinity ax-inf2 8705 requires this axiom along with Regularity ax-reg 8656 for its derivation (as theorem axinf2 8704 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2 8705 instead of this one. The derivation of this axiom from ax-inf2 8705 is shown by theorem axinf 8708.

Proofs should normally use the standard version ax-inf2 8705 instead of this axiom. (New usage is discouraged.) (Contributed by NM, 16-Aug-1993.)

Assertion
Ref Expression
ax-inf 𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))
Distinct variable group:   𝑥,𝑦,𝑧,𝑤

Detailed syntax breakdown of Axiom ax-inf
StepHypRef Expression
1 vx . . . 4 setvar 𝑥
2 vy . . . 4 setvar 𝑦
31, 2wel 2146 . . 3 wff 𝑥𝑦
4 vz . . . . . 6 setvar 𝑧
54, 2wel 2146 . . . . 5 wff 𝑧𝑦
6 vw . . . . . . . 8 setvar 𝑤
74, 6wel 2146 . . . . . . 7 wff 𝑧𝑤
86, 2wel 2146 . . . . . . 7 wff 𝑤𝑦
97, 8wa 382 . . . . . 6 wff (𝑧𝑤𝑤𝑦)
109, 6wex 1852 . . . . 5 wff 𝑤(𝑧𝑤𝑤𝑦)
115, 10wi 4 . . . 4 wff (𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦))
1211, 4wal 1629 . . 3 wff 𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦))
133, 12wa 382 . 2 wff (𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))
1413, 2wex 1852 1 wff 𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))
Colors of variables: wff setvar class
This axiom is referenced by:  zfinf  8703
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