Cantor's Non-Equinumerosity Theorems, Inductively

TL;DR

This work revisits Cantor's non-equinumerosity theorems by presenting inductive constructions that parallel Cantor's diagonal and anti-diagonal arguments. It proves the uncountability of the set of infinite binary sequences and of the real numbers via inductive constructions that ensure the constructed object differs from every item in any given list, with careful handling of base representations. The powerset theorem is extended through inductive proofs that rely on well-orderings (and hence the Axiom of Choice or equivalent principles), offering multiple inductive formulations and connecting them to standard Cantor-style arguments. The results highlight how induction, including transfinite induction and maximality arguments, can robustly establish non-surjectivity results, while clarifying the role of Choice in the powerset case.

Abstract

We apply an inductive argument to three theorems of Cantor on (1) the uncountability of infinite binary sequences, (2) the uncountability of real numbers, and (3) the non-equinumerosity of sets with their powersets. This technique proves the powerset theorem by assuming the Axiom of Choice.

Theorems & Definitions (6)

• Theorem 1.1: Cantor, 1890
• Theorem 2.1: Cantor, 1874
• Theorem 3.1: Powerset Theorem for Well-Ordered Sets
• Theorem 3.2: Cantor's Powerset Theorem, I
• Theorem 3.3: Cantor's Powerset Theorem, I I
• Proposition 3.4: $\mathcal{B}=\mathcal{D}_\infty$