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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.

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) …
Noah Schweber's user avatar
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, …
Noah Schweber's user avatar
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^ …
Noah Schweber's user avatar
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 …
Noah Schweber's user avatar
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 …
Noah Schweber's user avatar
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 …
Noah Schweber's user avatar
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 …
Noah Schweber's user avatar
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 …
Noah Schweber's user avatar