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Results tagged with continuum-hypothesis
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user 8133
Questions about the continuum hypothesis, or where the continuum hypothesis or its negation plays a role. This tag is also suitable, by extension, to refer to the generalized continuum hypothesis and related issues.
10
votes
Consistency of $c=2^{\aleph_0}=2^{\aleph_1}=\ldots=2^{\aleph_n}\ldots$, for every $n<\omega$
The answer is yes. Starting for simplicity with a model of $\mathsf{ZFC+GCH}$, if we force to add $\aleph_{\omega+1}$-many Cohen reals we will get (by the usual "nice names" analysis) $2^{\aleph_0}=2^ …
- 21.1k
15
votes
Accepted
Almost everywhere “filling” of the continuum by the first uncountable cardinal without CH
The existence of such an $f$ is consistent with $\mathsf{ZFC+\neg CH}$. Suppose $\mathfrak{c}=\aleph_2$ and every set of size $\le\aleph_1$ is null (this is consistent with $\mathsf{ZFC}$; it follows, …
- 21.1k
3
votes
Bernstein's proof of the continuum hypothesis
Ackermann's above-cited review focuses on the second of Bernstein's proposed rules. However, even the first ("axiom of identity") is broken. Gentling tweaking the language, this axiom reads:
$(\star) …
- 21.1k
8
votes
Very Large Cardinal Axioms and Continuum Hypothesis
Here's a brief sketch of why, assuming $ZFC+I_0$ is consistent, so is $ZFC+CH+I_0$. (This is just Levy-Solovay.)
Suppose $\lambda$ is $I_0$ - that is, there is a nontrivial elementary embedding $j$ o …
- 21.1k
4
votes
Accepted
What's the consistency status/strength of this limitation principle?
This principle is inconsistent: consider the formula $\theta(x)$ = "$x^+$ is the smallest infinite cardinal at which $\mathsf{CH}$ fails." The formula $\theta$ cannot hold on more than one infinite ca …
- 21.1k
2
votes
Accepted
Continuum function maximum
The intricacies of arithmetic at singular cardinals notwithstanding, I think you're looking for something which doesn't exist.
Given any cardinals $\kappa,\lambda$, singular or regular, there is a (se …
- 21.1k
9
votes
Mathematical Evidence Backing $|\mathbb{R}|=\aleph_2$
A partial answer to (2): my recollection is that the strongest viewpoint Godel ever put forth on the value of the continuum was that it should be $\aleph_2$, this being captured in his manuscript "Som …
- 21.1k
47
votes
Accepted
Hilbert's alleged proof of the Continuum Hypothesis in "On the Infinite"
The article "Hilbert and Set Theory" by Dreben and Kanamori devotes Section 7 to this argument and an analysis of its flaws. Dreben and Kanamori use the translation provided by van Heijenoort, so that …
- 21.1k