Counterexamples to Cantorian Set Theory

TL;DR

This paper challenges fundamental aspects of Cantorian Set Theory by providing counterexamples that question the validity of the Diagonal Method, the concept of set cardinality, and the non-denumerability of real numbers, proposing a new perspective based on representations.

Contribution

It introduces counterexamples to Cantor's theorems, redefines set cardinality based on representations, and proposes a model-based approach to set well-ordering.

Findings

Counterexample invalidates Cantor's Diagonal Method for reals.

Set cardinality depends on representation, not size.

Sets with different cardinalities can be equipollent.

Abstract

This paper provides some counterexamples to Cantor's contributions to the foundations of Set Theory. The first counterexample forces Cantor's Diagonal Method (DM) to yield one of the numbers in the target list. To study this anomaly, and given that for the DM to work the list of numbers have to be written down, the set of numbers that can be represented using positional fractional notation, $\mathbb{W}$, is properly characterized. It is then shown that $\mathbb{W}$ is not isomorphic to $\mathbb{R}$, meaning that results obtained from the application of the DM to $\mathbb{W}$ in order to derive properties of $\mathbb{R}$ are not valid. It is then shown that Cantor's DM for a generic list of reals can be forced to yield one of the numbers of the list, thus invalidating Cantor's result that infers the non-denumerability of $\mathbb{R}$ from the application of the DM to $\mathbb{W}$. Cantor's Theorem about the different cardinalities of a set and its power set is then questioned, and by means of another counterexample we show that the theorem does not actually hold for infinite sets. After analyzing all these results, it is shown that the current notion of cardinality for infinite sets does not depend on the size of the sets, but rather on the representation chosen for them. Following this line of thought, the concept of model as a framework for the construction of the representation of a set is introduced, and a theorem showing that an infinite set can be well-ordered if there is a proper model for it is proven. To reiterate that the cardinality of a set does not determine whether the set can be well-ordered, a set of cardinality $\aleph_{0}^{\aleph_{0}}$ is proven to be equipollent to the set of natural numbers $\mathbb{N}$. The paper concludes with an analysis of the cardinality of the ordinal numbers, for which a representation of cardinality $\aleph_{0}^{\aleph_{0}}$ is proposed.