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Lemma·L04

ω is the least inductive set

The naturals form an inductive set contained in every inductive set.

(i) is (D025); (ii) for every inductive set , where is the set of natural numbers.
In words
Omega is itself inductive, and it is contained in every inductive set: it is the smallest inductive set there is.
Rests onF09
Never needed: F02 · F03 · F04 · F05 · F06 · F08 · F10 · F11 · F12 · F13 · F14 · A01 · A02 · A03 · A04 · A05 · A06 · A07 · A08 (computed from the citation graph, not asserted).

Proof

  1. 1
    (ii) Let be inductive and . By D026, belongs to every inductive set, in particular . Hence by D001.
  2. 2
    (i) First, (D002) : every inductive set contains by the first clause of D025, so satisfies the defining condition of D026.
  3. 3
    Second, closure: let and let be any inductive set. Then by D026, so (D006) by the second clause of D025. As was arbitrary (universal generalization), belongs to every inductive set, i.e. . So is inductive.
∎

Remarks

The two parts together say is the least inductive set, which is the precise sense in which the naturals contain "zero, its successors, and nothing else". Everything about induction flows from this minimality: T05 is little more than part (ii) applied to a subset of .
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Used by

T05T06
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