Continuum Hypothesis- questions regarding proof/disproof

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Continuum Hypothesis- questions regarding proof/disproof

Postby xepher » Thu Jan 06, 2011 1:47 pm UTC

I understand what the continuum hypothesis is. It states that the cardinality of the numbers between zero and one is the same as the cardinality of numbers between one and two, or two and three, or three and four, etc.
Apparently this cannot be proven or disproven.
But it's SO FREAKING OBVIOUS (at least to me)!
Why in the world would it not be true? For each element of the numbers between zero and one, we just add a natural number to each, and we should have all the numbers between the natural number we chose and the one right above it.

Pardon me if there's something wrong here, but I just don't get how it's impossible to prove or disprove.

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Re: Continuum Hypothesis- questions regarding proof/disproof

Postby jaap » Thu Jan 06, 2011 2:19 pm UTC

What is wrong is that that is not at all what the continuum hypothesis says.

Consider these two sets:
- the set of natural numbers, {0, 1, 2, 3, ...}
- the set of reals between 0 and 1, the interval (0,1).

Both have infinitely many elements. Cantor proved however that they have a different cardinality, i.e. that the second set has in a sense more elements than the first. In other words that there are different types of infinity. It is possible to construct 'larger' infinities than these two as well.

The continuum hypothesis states that there are no infinities of a size between the two I mentioned, i.e. that there is no cardinality between the countable infinity of the natural numbers, and the infinity of the reals ("the continuum").

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Re: Continuum Hypothesis- questions regarding proof/disproof

Postby Yakk » Thu Jan 06, 2011 4:15 pm UTC

It (the Continuum Hypothesis, or CH) was proposed, then proven independent of the standard axioms (ZFC). ZFC+CH and ZFC+~CH has been studied (ZFC with the CH axiom, and ZFC with the negation of the CH axiom). With ZF+~CH, there are extra axioms you can add in to detail what kind of properties the sets with cardinality between N and R have (are they "natural number" like in some sense?) (ZFC is ZF set theory with the axiom of choice) (ZF set theory is Zermelo–Fraenkel set theory, one of the more popular (most popular?) formal axiomatic set theories in use today).

Edit: Removed a "-" from where it didn't belong.
Last edited by Yakk on Thu Jan 06, 2011 10:50 pm UTC, edited 1 time in total.
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Re: Continuum Hypothesis- questions regarding proof/disproof

Postby Eebster the Great » Thu Jan 06, 2011 10:39 pm UTC

ZF and ZFC are pretty much used everywhere.

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