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

Research output: Contribution to journalArticle

### 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 language English 39-66 27 Theoretical Mathematics & Applications 6 1 Published - Jan 2016

Axiom
Countable
Infinity
Cantor
Alternatives

### Keywords

• Axiomatic Set Theory
• Logic
• Foundation of Mathematics

### Cite this

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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",
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volume = "6",
pages = "39--66",
journal = "Theoretical Mathematics & Applications",
issn = "1792-9687",
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In: Theoretical Mathematics & Applications, Vol. 6, No. 1, 01.2016, p. 39-66.

Research output: Contribution to journalArticle

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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

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KW - Logic

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