# Principle of Mathematical Induction/Zero-Based

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## Theorem

Let $\map P n$ be a propositional function depending on $n \in \N$.

Suppose that:

- $(1): \quad \map P 0$ is true

- $(2): \quad \forall k \in \N: k \ge 0 : \map P k \implies \map P {k + 1}$

Then:

- $\map P n$ is true for all $n \in \N$.

## Proof

Consider $\N$ defined as a Peano structure.

The result follows from Principle of Mathematical Induction for Peano Structure.

$\blacksquare$

## Also see

- Results about
**Proofs by Induction**can be found here.

## Sources

- 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $2$: Integers and natural numbers: $\S 2.1$: The integers - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): Appendix $\text{A}.12$: Induction

- 2008: Paul Halmos and Steven Givant:
*Introduction to Boolean Algebras*... (previous) ... (next): Appendix $\text{A}$: Set Theory: Induction - 2012: M. Ben-Ari:
*Mathematical Logic for Computer Science*(3rd ed.) ... (previous): Appendix $\text{A}.6$