Talk:Continuum hypothesis

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Equivalent?

@Maeyn: deleted an alleged equivalent of the continuum hypothesis which he claimed is not actually equivalent. The continuum hypothesis says (A) "There is no set whose cardinality is strictly between that of the integers and the real numbers.". The alleged equivalent says (B) "Any subset of the real numbers is either finite, or countably infinite, or has the cardinality of the real numbers.". ~~I believe that it is, in fact, actually equivalent.~~ JRSpriggs (talk) 16:52, 19 September 2025 (UTC)[reply]

At least if you have countable choice for reals, yes, that is equivalent. Without a tiny bit of choice, maybe it's slightly stronger, since it excludes an infinite but Dedekind-finite set of reals? Not sure whether or not there are models with such a thing. But we're assuming choice in the background in this article, as in mathematics generally.

That said, I don't really see why we should call that long-winded version out, particularly with the separation between "finite" and "countably infinite", which are normally lumped together as "countable". (There is a minority usage that does not consider finite sets to be countable but that could be mentioned in an explanatory footnote; we don't have to break up the flow with it.) I wouldn't be against mentioning "every set of reals is either countable or has the cardinality of the continuum" as an equivalent, if we don't already. --Trovatore (talk) 18:24, 19 September 2025 (UTC)[reply]