The notion of Infinity within the Zermelo system and its relation to the Axiom of Countable Choice

Research output: Contribution to journalArticle

2 Downloads (Pure)

Abstract

In this article we consider alternative definitions-descriptions of a set being Infinite within theprimitive Axiomatic System of Zermelo, Z . We prove that in this system the definitions ofsets being Dedekind Infinite, Cantor Infinite and Cardinal infinite are equivalent each other.
Additionally, we show that assuming the Axiom of Countable Choice,
these definitions are also equivalent to the definition of a set being Standard Infinite, that is, of not being finite. Furthermore, we show that Dedekind infinitness is weaker than Axiom of Countable Choice
Original languageEnglish
Pages (from-to)39-66
Number of pages27
JournalTheoretical Mathematics & Applications
Volume6
Issue number1
Publication statusPublished - Jan 2016

Fingerprint

Axiom
Countable
Infinity
Cantor
Alternatives

Keywords

  • Axiomatic Set Theory
  • Logic
  • Foundation of Mathematics

Cite this

@article{055ca49a930d48558ba6e5d0e6f41f35,
title = "The notion of Infinity within the Zermelo system and its relation to the Axiom of Countable Choice",
abstract = "In this article we consider alternative definitions-descriptions of a set being Infinite within theprimitive Axiomatic System of Zermelo, Z . We prove that in this system the definitions ofsets being Dedekind Infinite, Cantor Infinite and Cardinal infinite are equivalent each other.Additionally, we show that assuming the Axiom of Countable Choice,these definitions are also equivalent to the definition of a set being Standard Infinite, that is, of not being finite. Furthermore, we show that Dedekind infinitness is weaker than Axiom of Countable Choice",
keywords = "Axiomatic Set Theory, Logic, Foundation of Mathematics",
author = "George Chailos",
year = "2016",
month = "1",
language = "English",
volume = "6",
pages = "39--66",
journal = "Theoretical Mathematics & Applications",
issn = "1792-9687",
number = "1",

}

TY - JOUR

T1 - The notion of Infinity within the Zermelo system and its relation to the Axiom of Countable Choice

AU - Chailos, George

PY - 2016/1

Y1 - 2016/1

N2 - In this article we consider alternative definitions-descriptions of a set being Infinite within theprimitive Axiomatic System of Zermelo, Z . We prove that in this system the definitions ofsets being Dedekind Infinite, Cantor Infinite and Cardinal infinite are equivalent each other.Additionally, we show that assuming the Axiom of Countable Choice,these definitions are also equivalent to the definition of a set being Standard Infinite, that is, of not being finite. Furthermore, we show that Dedekind infinitness is weaker than Axiom of Countable Choice

AB - In this article we consider alternative definitions-descriptions of a set being Infinite within theprimitive Axiomatic System of Zermelo, Z . We prove that in this system the definitions ofsets being Dedekind Infinite, Cantor Infinite and Cardinal infinite are equivalent each other.Additionally, we show that assuming the Axiom of Countable Choice,these definitions are also equivalent to the definition of a set being Standard Infinite, that is, of not being finite. Furthermore, we show that Dedekind infinitness is weaker than Axiom of Countable Choice

KW - Axiomatic Set Theory

KW - Logic

KW - Foundation of Mathematics

M3 - Article

VL - 6

SP - 39

EP - 66

JO - Theoretical Mathematics & Applications

JF - Theoretical Mathematics & Applications

SN - 1792-9687

IS - 1

ER -