Induction

Pub Mar 06, 2026 Up Jun 10, 2026

Proposition 1. Induction. Let $A$ be a subset of $\N$. Suppose:

  1. \[0 \in A.\]
  2. \[\forall n \in A : n + 1 \in A.\]

Then $A = \N$.

Proof

Otherwise, $\min(\N \setdif A)$ would be a limit ordinal, contrary to FINO > Proposition 1.

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References

[1]
Wikipedia contributors. Mathematical induction. Wikipedia, The Free Encyclopedia.
[2]
Thomas Jech. Set Theory. 3rd Millennium ed, rev. and expanded. Springer, 2002. p. 21.