Inconsistency of uncountable infinite sets under ZFC framework

The purpose is to show that all uncountable infinite sets are self‐contradictory non‐sets.

A conceptual approach is taken in the paper.

Given the fact that the set N={x|n(x)} of all natural numbers, where n(x)=_df “x is a natural number” is a self‐contradicting non‐set in this paper, the authors prove that in the framework of modern axiomatic set theory ZFC, various uncountable infinite sets are either non‐existent or self‐contradicting non‐sets. Therefore, it can be astonishingly concluded that in both the naive set theory or the modern axiomatic set theory, if any of the actual infinite sets exists, it must be a self‐contradicting non‐set.

The first time in history, it is shown that such convenient notion as the set of all real numbers needs to be reconsidered.