Unifying some notions of infnity in ZC and ZF

Let ZC -  I (respectively, ZF -  I) be the theory obtained by deleting the axiom of infinity from the usual list of axioms for Zermelo set theory with choice (respectively, the usual list of axioms for Zermelo-Fraenkel set theory). In this note, we present a collection of sentences 9x'(x) for which (ZC -  I) + 9x'(x) (respectively, (ZF - I)+9x'(x)) proves the existence of an infinite set.

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