What question did this study set out to answer?

This paper aims to explore the distinctions between bijections in infinite sets and highlight Cantor's logical fallacies.

July 10, 2026Open Access

Two Types of Bijections and Cantor's Logical Fallacy

Key Points

This paper aims to explore the distinctions between bijections in infinite sets and highlight Cantor's logical fallacies.
Logical analysis of bijections between infinite sets.
Comparison of partial and full field bijections.
Evaluating Cantor's conclusions on cardinalities.
Cantor's confusions between partial and full field bijections are clarified.
The erroneous conclusion |N| < |P(N)| is substantiated as a logical fallacy.
It is demonstrated that |R| = |P(N)| contains flaws.

Abstract

There are two types of bijections between infinite sets: partial field bijection and full field bijections. The bijections used to compare the cardinalities of sets are partial field in nature, exemplified by the bijection between even numbers and natural numbers. Cantor failed to distinguish between these two different types of bijections, which led to logical fallacies and erroneous conclusions, such as |N| < |P(N)| and |R| = |P(N)|. This paper logically demonstrates Cantor's logical errors.

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