Talk:Continuum hypothesis
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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
Under CH, there exists bounded functions from the unit square into R, which are measurable in each coordinate, but so that the functions is not integrable. It is consistent with ZFC that no such function exists. (So CH is necessary.) Although if you permute any of the conditions slightly, the situation can be resolved in ZFC.
There are plenty of things known to be independent of ZFC but which don't line up so nicely with CH. (Kaplansky's problem and the Whitehead problem are two good examples.) --
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
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?
Is Chris Freiling's "statement about probabilities" Freiling's Axiom of Symmetry? If so, somebody should add a link.
techniccalyy there aren fractal dimensions that are applicableto set theory and cardinallity that raise questions as to whether Cantors statment
'There is no set whose size is strictly between that of the integers and that of the real numbers.'
is correct
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Removed statement that it would be impossible to prove that ZF contains a contradiction.
Roadrunner 21:59, 20 Apr 2004 (UTC)
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I've been told that logicians now have reason to think that the "continuum hypothesis is false." Apparently the situation is that people basically want to take the axiom of projective determinancy, and it's now been shown that the axiom of projective determinacy impiles the negation of the contiuum hypothesis. I'll have to leave editing the article to someone who actually knows what's going on.
I found a relevant link: http://math.berkeley.edu/~woodin/talks/Lectures.html
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Absolutely, Woodin's work is relevant, and should be referred to somewhere in the article. Unfortunately it's rather hard to summarize in a way that would make much sense to more than, at a generous estimate, 1000 people in the world.
Projective determinacy does not in fact imply the negation of CH (ZFC+PD+CH is consistent, assuming ZFC+PD is itself consistent). Rather, Woodin offers PD as an example of a proposition that has a kind of indestructibility via forcing. Woodin proposes an allegedly similar sort of stability under forcing for the theory of (I think) H(ω2), and argues that this stability implies ~CH. Or something like that. I'm pretty fuzzy on the details myself. --Trovatore 00:43, 26 Jun 2005 (UTC)
So I didn't try to explain it, just added the ref and pointed the reader to it. Trovatore 05:04, 27 Jun 2005 (UTC)
[edit] Change hyperlink?
Shouldn't "Zermelo-Fränkel set theory axiom system" be a hyperlink to Zermelo-Fraenkel set theory instead of Axiomatic set theory?
[edit] who believes CH or ~CH ?
I don't entirely agree with the following passage:
Generally speaking, mathematicians who favor a "rich" and "large" universe of sets are against CH, while those favoring a "neat" and "controllable" universe favor CH.
See the Maddy reference I added, page 500, the sections entitled Not-CH is restrictive (in favor) and Modern forcing (in favor).
I'll try to come up with some wording that's not too awkward, illustrating the historical view reported in the existing Wiki article, while also pointing out that between models having all the same reals, it's the ones with more sets of reals that are more likely to satisfy CH. --trovatore
My attempt is now in place Trovatore 05:04, 27 Jun 2005 (UTC)
[edit] Investigating the continuum hypothesis
I've removed this section:
- If a set S were found that disproved the continuum hypothesis, it would be impossible to make a one-to-one correspondence between S and the set of integers, because there would always be elements of set S that were "left over". Similarly, it would be impossible to make a one-to-one correspondence between S and the set of real numbers, because there would always be real numbers that were "left over".
It's a good thing to add some intuition, but I don't know that this passage helped much. CH isn't about whether such a set S can be found in any ordinary sense, but just about whether one exists. --Trovatore 21:12, 22 October 2005 (UTC)
[edit] Simplifying the language a bit
Quoting from the Manual of style:
Do not assume that your reader is familiar with the acronym or abbreviation you are using. The standard writing style is to spell out the acronym or abbreviation on the first reference (wikilinked if appropriate) and then show the acronym or abbreviation after it.
I say this because it took me about 10 seconds to realise that in ZF + GCH ⊦AC, AC stood for Axiom of Choice. I daresay there are a number of people who want to understand this article but wouldn't even know what ⊦ means (especially if they don't know to check an article about formal logic). Perhaps the first reference to Zemillo-Frankael should include (ZF) after it, and likewise include a brief statement somewhere that ZF + Axiom of Choice (AC) = ZFC. Confusing Manifestation 02:26, 24 February 2006 (UTC)
