Theoretical Mathematics & Applications

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

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•  Abstract

  In this article we consider alternative definitions-descriptions of a set being Infinite within the primitive Axiomatic System of Zermelo, Z. We prove that in this system the definitions of sets being Dedekind Infinite, Cantor Infinite and Cardinal infinite are equivalent each other. Additionally, we show that assuming the Axiom of Countable Choice, AC_no, these definitions are also equivalent to the definition of a set being Standard Infinite, that is, of not being finite. Furthermore, we consider the relation of AC_no (and some of its special cases) with the statement“A set is Standard Infinite if and only if it is Dedekind Infinite”. Among other results we show that the system Z+SD is ‘strictly weaker’ than Z+AC_n0 .