Principle of Finite Induction/Zero-Based

Theorem

Let $S \subseteq \N$ be a subset of the natural numbers.


Suppose that:

$(1): \quad 0 \in S$
$(2): \quad \forall n \in \N : n \in S \implies n + 1 \in S$


Then:

$S = \N$


Proof

Consider $\N$ defined as a naturally ordered semigroup.

The result follows directly from Principle of Mathematical Induction for Naturally Ordered Semigroup.

$\blacksquare$


Sources

  • 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 4$: Number systems $\text{I}$: Peano's Axioms
  • 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): Appendix $\text{A}$: Set Theory: Induction
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