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A '''proof by induction''' is a technique of mathematical [[proof]]. It is one of the axioms of [[Zermelo-Fraenkel]] set theory, allowing a mathematician to have proofs that count in order through all the [[Natural_number|natural numbers]]. Proofs by induction come in three flavors: weak (or classic) induction, strong induction, and transfinite induction. As the names suggest, weak induction proofs require fewer assumptions than strong induction proofs. Sometimes these assumptions are too weak, in which case strong induction is necessary. Transfinite induction involves infinitary mathematics, including the [[Axiom of Choice]]. Many mathematicians avoid transfinite induction when possible.

A '''proof by induction''' is a technique of mathematical [[proof]]. It is one of the axioms of [[Zermelo-Fraenkel]] set theory, allowing a mathematician to have proofs that proceed in order through all the [[Natural_number|natural numbers]]. Proofs by induction are divided into three categories: weak (or classic) induction, strong induction, and transfinite induction. As the names suggest, weak induction proofs require fewer assumptions than strong induction proofs. Sometimes these assumptions are too weak, in which case strong induction is necessary. Transfinite induction involves infinitary mathematics, including the [[Axiom of Choice]]. Many mathematicians avoid transfinite induction when possible.

=== Weak Induction ===

=== Weak Induction ===