This is a question of terminological convention rather than mathematics. Does "large cardinal" refer specifically to something in the context of ZF/ZFC, or can it be used for other theories (in particular Zermelo set theory Z, which is ZF without the axiom of replacement)? If I understand correctly, Z doesn't prove the existence of ${\displaystyle \aleph _{\omega +\omega }}$, so I'm asking whether this cardinal would be considered "large" in Z in some reasonably normal terminology (not that anyone nowadays talks about Z that much in the first place). Thanks. 69.228.171.139 (talk) 07:45, 13 June 2012 (UTC)[reply]

I don't know if there's any source that refers to the existence of ${\displaystyle \aleph _{\omega +\omega }}$as a "large-cardinal axiom", but it seems reasonable to me. John R. Steel says that large-cardinal axioms are "natural markers of consistency strength", which would certainly seem to fit: Over Z, the existence of ${\displaystyle \aleph _{\omega +\omega }}$proves Con(Z) and more, in a very natural way. --Trovatore (talk) 08:28, 13 June 2012 (UTC)[reply]