Principle of Finite Induction

Theorem
Let $S \subseteq \Z$ be a subset of the integers.

Let $n_0 \in \Z$ be given.

Suppose that:


 * $(1): \quad n_0 \in S$


 * $(2): \quad \forall n \ge n_0: n \in S \implies n + 1 \in S$

Then:


 * $\forall n \ge n_0$: $n \in S$

That is:
 * $S = \set {n \in \Z: n \ge n_0}$

The principle of finite induction is usually stated and demonstrated for $n_0$ being either $0$ or $1$.

This is often dependent upon whether the analysis of the fundamentals of mathematical logic are zero-based or one-based.

Contexts
The Principle of Finite Induction can be introduced in a formal development of abstract algebra or mathematical logic in various contexts, and proved from first principles in each.

Also known as
The Principle of Finite Induction is often referred to as the principle of mathematical induction, but the latter is usually reserved for a slightly different concept.

Also see

 * Principle of Mathematical Induction


 * Second Principle of Finite Induction
 * Second Principle of Mathematical Induction