Inconsistency of uncountable infinite sets under ZFC framework

Inconsistency of uncountable infinite sets under ZFC framework Purpose – The purpose is to show that all uncountable infinite sets are self‐contradictory non‐sets. Design/methodology/approach – A conceptual approach is taken in the paper. Findings – 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. Originality/value – The first time in history, it is shown that such convenient notion as the set of all real numbers needs to be reconsidered. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Kybernetes Emerald Publishing

Inconsistency of uncountable infinite sets under ZFC framework

Kybernetes, Volume 37 (3/4): 5 – Apr 11, 2008

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Publisher
Emerald Publishing
Copyright
Copyright © 2008 Emerald Group Publishing Limited. All rights reserved.
ISSN
0368-492X
DOI
10.1108/03684920810863417
Publisher site
See Article on Publisher Site

Abstract

Purpose – The purpose is to show that all uncountable infinite sets are self‐contradictory non‐sets. Design/methodology/approach – A conceptual approach is taken in the paper. Findings – 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. Originality/value – The first time in history, it is shown that such convenient notion as the set of all real numbers needs to be reconsidered.

Journal

KybernetesEmerald Publishing

Published: Apr 11, 2008

Keywords: Cybernetics; Set theory; Number theory

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