Mathematical induction, transfinite induction, and induction over the continuum

Abstract

This article examines three types of induction methods in mathematics: mathematical induction, transfinite induction, and induction over the continuum. If a statement holds true for all natural numbers, it is proven using mathematical induction. If a statement holds true for all ordinal numbers, it is proven using transfinite induction. Since induction over the continuum cannot be applied to a statement, when something is said to be proven true for every point in [a, b), the proof is done using induction over the continuum.