Talk:Strongly inaccessible cardinal

Rename or merge

By default, "inaccessible" means "strongly inaccessible". This page should be moved to "Inaccessible cardinal", and the latter should be moved to "Weakly inaccessible cardinal". Or better, the two pages should be merged. --Trovatore 01:42, 14 July 2005 (UTC)[reply]

Yes, one inaccessible cardinal page is enough. Charles Matthews 08:25, 14 July 2005 (UTC)[reply]

I don't think so. The two concepts are not the same, thus they should both have their own articles. Moving this page to Inaccessible cardinal and renaming that one to Weakly inaccessible cardinal is fine with me, though. -- Schnee (cheeks clone) 22:53, 15 July 2005 (UTC)[reply]

Actually, I merged them boldly a couple days ago; Weakly inaccessible cardinal and Strongly inaccessible cardinal are now both redirects to Inaccessible cardinal. Nothing that can't be undone of course. But I'd urge against it strongly, for the following reasons:

The definitions of the concepts are closely analogous, differing only in "limit" versus "strong limit". "Weakly inaccessible" is a sort of lightface analogue of "inaccessible".
The properties occupy exactly the same rung of the consistency-strength ladder. In fact, for any ordinal α, there's a transitive model with α inaccessibles if and only if there's one with α weak inaccessibles, and the proof is trivial: Cut down to the L of the model, and the weak inaccessibles become strong.
Of course there's a lot more to large cardinals in general than their consistency strength, but in the case of weak inaccessibles, what else is there? The only thing I can really think of to say about them is that it's consistent that the cardinality of the continuum is weakly inaccessible, which would imply that it's a fixed point of the ℵ function in a strong way--but that note would fit nicely in the current article, or in the continuum hypothesis article.--Trovatore 00:24, 16 July 2005 (UTC)[reply]

By the way, the merged article still has one clear flaw (not counting possible expansions that haven't been done): The paragraph that says you can't prove weak inaccessibles exist in ZFC, and the one that says the same thing for strong inaccessibles, should really be merged somehow, especially in light of the equiconsistency fact I noted above.--Trovatore 00:27, 16 July 2005 (UTC)[reply]

Mmmm - you guys should have some mercy. I think the merge is fine. I would suggest quite a substantial expansion. The large cardinals pages tend to suffer from lack of motivation and history (which can always help); and from relentless technical density. Charles Matthews 08:44, 16 July 2005 (UTC)[reply]