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Axiom·A05

Infinity

There exists a set containing the empty set and closed under successors.

Here is D002 and is the successor of D006.
In words
There is a set I such that the empty set belongs to I and for every x, if x belongs to I then its successor S(x) belongs to I.
This is a ZFC axiom: it is assumed, not proven. Everything below it in a proof chain ultimately rests here.

Remarks

Asserts that an inductive set exists: a set containing and closed under the successor . Any such is infinite, since it hosts the distinct finite ordinals , and is the home of the natural numbers. The existence clause of this axiom (just " ") is also what feeds the empty-set derivation T01 together with Separation. This is the one axiom that brings the infinite into mathematics. In later terminology, such an is exactly an inductive set; the least one is ω, the set of natural numbers (L04).
Wikipedia ↗instInfiniteNat ↗

Used by

T01
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