# Questions tagged [continuum-hypothesis]

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.

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### Uniformization of almost disjoint families

Suppose $\mathcal{F} \subseteq \mathcal{P} (\omega)$ is an almost disjoint family and $\aleph_0 < \vert \mathcal{F} \vert = \kappa < 2^{\aleph_0}$. Is it consistent that for some such cardinal ...
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### Hereditarily Lindelöf spaces with density continuum

Since there are L-spaces (provably in ZFC), under CH we have regular, hereditarily Lindelöf spaces with density continuum. However, I cannot find an example of such a space under not CH, nor a proof ...
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### Bernstein's proof of the continuum hypothesis

In the paper The Continuumproblem, Felix Bernstein introduces a new axiom and uses it to conclude the continuum hypothesis. (1) As the paper is relatively old and the writing style is somehow informal,...
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### About the relationship between the generalized continuum hypothesis and the axiom of choice

I was trying to get a short, intuitive proof of Sierpinski’s theorem (gch implies axiom of choice) and I could but only by using the following gch2 for the generalized continuum hypothesis gch. gch: ...
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### Consistency of $c=2^{\aleph_0}=2^{\aleph_1}=\ldots=2^{\aleph_n}\ldots$, for every $n<\omega$

It is well known that the continuum cannot be $\aleph_\omega$. Instead, it can obtain any value in the aleph sequence up to that. My question is if it is consistent that all the powersets of the ...
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### What's the consistency status/strength of this limitation principle?

$\DeclareMathOperator\iCard{iCard}$In a prior posting If we limit matters what ZFC can prove, would that be consistent? to MO, I tried to capture the informal principle of whatever ZFC proves, it is, ...
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### Consistency of Generalised Continuum Hypothesis and univalence in HoTT

In homotopy type theory, propositional excluded middle and the axiom of choice sets are both consistent with univalence, both of which yields type theoretic models for classical mathematics. However, ... 2k views

### Unnecessary uses of the Continuum Hypothesis

This question was inspired by the MathOverflow question "Unnecessary uses of the axiom of choice". I want to know of statements in ZFC that can be proven by assuming the Continuum Hypothesis,...
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### Must uncountable standard models of ZFC satisfy CH?

In Cohen's article, The Discovery of Forcing, he says that "one cannot prove the existence of any uncountable standard model in which AC holds, and CH is false," and offers the following ...
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### If GCH is breached the same way before a singular of uncountable cofinality, would that breach extend to that singular?

By 1 step breach of the GCH I mean the following: $$2^{\aleph_{\alpha}} = \aleph_{\alpha+2}$$ Now, it is known that there are more constrains on the cardinality of power sets at singlular cardinals ...
1 vote
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### Can GCH fail everywhere in every finite way?

Since the $\sf GCH$ cannot fail everywhere everyway (see here), the question here is if it can fail everywhere in every finite manner, that if we have a strictly increasing function $f$ on the ...
1 vote
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### Is existence of a cardinal that witness non-failure of GCH everywhere everyway, a theorem of ZF?

In an earlier positing to $\mathcal MO$, it appears that the answer to if the $\sf GCH$ can fail everywhere in every way is to the negative, this is the case in $\sf ZFC$, however it also appears that ...
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### Can GCH fail everywhere every way?

The following question is about if it is compatible to add to $\sf ZF$ an axiom asserting the existence of a countable transitive model of $\sf ZF$ such that for every strictly increasing function $f$ ...
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### If someone can prove Goldbach conjecture assuming the continuum hypothesis, do we consider the conjecture proved?

If someone can prove Goldbach conjecture assuming the continuum hypothesis, do we consider the Goldbach conjecture proved? If ZFC+CH implies Goldbach, and if the Goldbach turn out to be false, then it ...
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### Continuum function maximum

Easton's theorem can give a very weak nontrivial constraint on continuum function, but it does not hold for singular cardinals. So: What are the non-trivial constraints on continuum function in ...
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### Does GCH for alephs imply the axiom of choice?

GCH for alephs means the statement that, for any aleph $\kappa$, there are no cardinals $\mathfrak{r}$ such that $\kappa<\mathfrak{r}<2^\kappa$. Does GCH for alephs imply the axiom of choice? ...
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### Are there any important geometric consequences of the Generalised Continuum Hypothesis?

In a paper by Rafael Dahman on the embedding of the long line into weakly complete vector spaces, those of the form $R^I$ for arbitrary $I$, and equipped with the Michal-Bastiani calculus, he notes a ...
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### Continuum hypothesis and cardinality of infinite tree paths [closed]

Suppose a tree with nodes located at levels $1,2,3...$. At each level the nodes branch into several nodes or do not branch. Does the cardinality of the set of all infinite paths in this tree depend on ...
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### $\Sigma^2_2$ absoluteness and $\diamondsuit$

This question touches on recent questions concerning the role of $\diamondsuit$ and the Continuum Hypothesis. In particular, I would like to know more information about a conjecture of Woodin on the ...
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### What if $\mathbb{R}$ is in bijection with the cardinals less than $\frak{c}$?

I was wondering whether it is consistent to have $\frak{c} = \aleph_{\frak{c}}$ where $\frak{c} = 2^{\aleph_0}$ is the cardinality of the reals (over ZFC). If so, what interesting consequences of this ...
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### Embeddability into $\beta\omega$ and $\omega^*$

It is well known that under CH every totally-disconnected compact F-space of weight at most $\omega_1$ can be embedded into the remainder $\omega^*=\beta\omega\setminus\omega$ of the Cech-Stone ...
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In his well-known paper Bellamy constructs an indecomposable continua with exactly two composants. The setup is as follows: We have an inverse-system $\{X(\alpha); f^\alpha_\beta: \beta,\alpha < \... 20 votes 1 answer 1k views ### On Erdös–Kakutani like Equivalents of the Failure of Continuum Hypothesis Among all mysterious equivalents of the Continuum Hypothesis and its negation, there is an algebraic combinatorial equivalent of$\neg \mathit{CH}$in Erdös and Kakutani - On non-denumerable graphs (... 14 votes 1 answer 573 views ### Continuum Hypothesis and the fact that every co-finite topological space, with uncountable underlying set , is contractible Let$X$be a co-finite topological space. If$|X| \ge 2^{\aleph_0}=\mathfrak c$, then$X$is contractible (https://en.wikipedia.org/wiki/Contractible_space) . Indeed, there is a bijection$f: X \times ... 267 views

### A strange planar set and the Continuum Hypothesis

Call a number abnormal if its decimal expansion doesn't feature every digit an infinite number of times. Call a triangle in ${\Bbb R}^2$ abnormal if at least one of its angles spans an abnormal ...
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### Is $[0,1]$ a disjoint union of $\aleph_1$ compact subsets with empty interior?

Is $[0,1]$ a disjoint union of $\aleph_1$ compact subsets with empty interior? The answer is obviously yes assuming the continuum hypothesis. Also, by Baire's lemma, the answer is negative if one ...
### Is it possible to define cardinals that are distinct from either the $\aleph$ numbers or $\beth$ numbers?
I am wondering if there are ways of defining "structure" on infinite sets that generate sequences of cardinals that cannot be proved to have the same cardinality as either the $\aleph$ or $\beth$ ...