### Abstract

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 |
---|---|

Pages (from-to) | 39-66 |

Number of pages | 27 |

Journal | Theoretical Mathematics & Applications |

Volume | 6 |

Issue number | 1 |

Publication status | Published - Jan 2016 |

### Fingerprint

### Keywords

- Axiomatic Set Theory
- Logic
- Foundation of Mathematics

### Cite this

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*Theoretical Mathematics & Applications*, vol. 6, no. 1, pp. 39-66.

**The notion of Infinity within the Zermelo system and its relation to the Axiom of Countable Choice.** / Chailos, George.

Research output: Contribution to journal › Article

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 -