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, ACno, 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 ACno (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+ACn0 .