Talk:Continuum hypothesis

Are there any "practical consequences" of presuming CH (or GCH) to be true or false? For example, are there any theorems which can be proved by presuming CH to be true or by presuming it to be false, but for which the proof is much simpler if we presume CH to be true, (or vice versa). -- SJK

In set theory and analysis, there are many statements which could be proven if GCH is assumed, and could be disproven if GCH fails. I don't have any examples right now. I don't know if there are theorems whose proofs get simpler by assuming GCH. --AxelBoldt

There are. I don't remember any examples, but I distinctly remember moaning "if only we could just use GCH..." doing a proof on an assignment. --

I don't remember ever needing the continuum hypothesis. A list of examples is sorely needed in the article, otherwise the paragraph about "substantial results" should be deleted. Also, since the GCH implies the axiom of choice, it is much more likely that just the axiom of choice would suffice. -- Miguel

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Gödel's incompleteness theorems only say that if proof is identified with first-order logical derivation, then any consistent axiomatization will be incomplete. But his proof of the first theorem has two parts: the first proves that his wff U is unprovable; the second gives a proof of U (or rather its interpretation in N). The statement, "This statement is not first-order derivable from the given axioms" is surely provable, though not first-order derivable.

Likewise, CH has not been shown unprovable, but only underivable from ZFC.

Chris Freiling's "Axioms of Symmetry: Throwing Darts at the Real Number Line" (Journal of Symbolic Logic Vol. 51, Iss. 1, pp. 190-200) presents a (rather philosophical) argument against CH. --Archibald Fitzchesterfield

Yes, that's a good paper, I'll add it to the list of references. --AxelBoldt

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I find this article ot be hard to understand. I had to go through several other articles to even understand what it was about. Maybe someone should add a short informal summary that explains what the continuum hypothesis means that is also easier to understand. Right now I'd guess that if you understand the article you already know what the continuum hypothesis is. The article then loses a big part of its use. -XeoX

better?