Set Theory and Logic

Cantor's Continuum Hypothesis Is Proved Wrong

Authors: Chang Hee Kim
Comments: 22 Pages. CC BY 4.0

This paper demonstrates that Cantor’s Continuum Hypothesis is fundamentally flawed. The argument begins by showing that the set M = {0} U N can be decomposed into infinitely many subsets, each of which is infinite and pairwise disjoint from the other subsets of M. These subsets are then shown to admit a one-to-one correspondence (bijection) with the unit interval [0, 1).We further examine the failure of Cantor’s Diagonal Argument, specifically exposing the structure of the implicit matrix it relies on. By decomposing M into such disjoint infinite subsets, we construct a direct bijection to the rows (real decimals) and columns (decimal digits) of this matrix. Each real number in [0, 1) corresponds uniquely to one of these subsets, eliminating the need to invoke "uncountable" sets.Through this decomposition, we establish that all infinite sets are equal. The very notion of comparing sizes of infinite sets — so-called "cardinality" — becomes unnecessary. As a result, the foundation of the Continuum Hypothesis is no longer valid.
Category: Set Theory and Logic