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
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 |
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Pages (from-to) | 39-66 |
Number of pages | 27 |
Journal | Theoretical Mathematics & Applications |
Volume | 6 |
Issue number | 1 |
Publication status | Published - Jan 2016 |
Keywords
- Axiomatic Set Theory
- Logic
- Foundation of Mathematics