# 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 ...

13
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1
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### Almost everywhere “filling” of the continuum by the first uncountable cardinal without CH

Assuming the negation of CH, let $\omega_1$ be the first uncountable ordinal, $\mathfrak{c}$ be the cardinality of the continuum. Does there exist a map $f: \omega_1 \times [0, 1] \rightarrow \...

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### Can the cardinal $2^{\aleph_0}$ be order-embedded in ${\cal P}(\omega)/(\text{fin})$?

For $A,B\in{\cal P}(\omega)$ we say $A\subseteq^* B$ if $A\setminus B$ is finite (that is, $A$ is "almost contained" in $B$). We write $A\simeq_{\text{fin}} B$ if $A\subseteq^* B$ and $B\...

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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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1
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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: ...

7
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1
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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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### Weak form of $\text{CH}$ in $L(\mathbb{R})$

I was wandering whether this weak form of $\text{CH}$ holds in $L(\mathbb{R})$ provably in $\text{ZF}+\text{DC}$
$(\text{ZF}+\text{DC}) \ L(\mathbb{R})\vDash \forall X\subseteq\mathbb{R} ( X \text{ ...

2
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0
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### Weak form of CH in $L(\mathbb{R})$, reference

I've heard someone saying that in $L(\mathbb{R})$ the following form of $\text{CH}$ holds:
$L(\mathbb{R})\vDash \forall X\subseteq \mathbb{R}(X \text{ countable or } \mathbb{R}\le^* X)$, i.e. every ...

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### Given a partition of a field, construct a partition of its extension

The motivation for my question is the following algebraic consequence of the Continuum Hypothesis ($2^{\aleph_0}=\aleph_1$) by Zoli:
(T1) Assume the Continuum Hypothesis holds. Then $\mathbb{R}^{\...

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1
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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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### What is the consistency strength of the following pattern of failure of the continuum hypothesis?

What is the least theory in which the following sentence is proved?
$ \exists M: M\text { is CTM(ZFC+ GCH)} \land \forall \kappa \in Card^M (\kappa > 1 \implies \\\exists N: N \text { is CTM(ZFC) }...

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### Which extension of ZFC proves that ZFC can only prove CH satisfied by the first two sets?

Which extension of $\sf ZFC$ prove that
$$ {\sf ZFC} \not \vdash \exists x \, ( \operatorname {CH}(x) \land x \neq \emptyset \land x \neq 1)$$
Where $\operatorname {CH}(x) \iff \neg \exists \kappa \, (...

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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, ...

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### 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 ...

2
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1
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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 ...

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0
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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 ...

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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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0
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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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0
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### PFA for cardinal preserving forcing notions and the CH

Let $FA_{\aleph_1}$(cardinal preserving proper forcings) be the forcing axiom: if $\mathbb{P}$ is a cardinal preserving proper forcing notion and $(D_\xi)_{\xi<\omega_1}$ are dense subsets of $\...

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2
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### Foundational results dependent on/equivalent to the continuum hypothesis or its negation?

I remember at a certain point early in my mathematical studies learning that the Axiom of Choice is equivalent to the following statement on Cartesian products:
If $\{ X_i \}_{i \in I}$ is any ...

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### Are there analogues of real-valued measurability for larger powersets?

Apologies if the question is somewhat naïve - I'm not a set theorist, just an enthusiast.
One possible intuition we could have about the generalized continuum hypothesis is that power sets are so ...

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### What's the consistency strength of resemblance + global failure of the continuum hypothesis?

Let $T$ be a theory formalized in first order logic with equality and membership and the additional primitive constant symbol $W$, with the following axioms:
Extensionality: $\forall z (z \in x \...

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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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### On the role of $\diamondsuit$

The well-known axiom $\diamondsuit$ states that there is a sequence $\langle A_\alpha:\alpha<\omega_1\rangle$ (a $\diamondsuit$-sequence) of countable sets with the property that for any $A\...

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### $\Sigma^2_1$ and the Continuum Hypothesis

This is a follow up to Will Brian's answer to this recent question. In particular, quoting Brian:
"In fact, Paul Larson has pointed out to me that the statement "$\phi$ and $\phi^{-1}$ are conjugate"...

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1
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### Can $H_{\omega_1}$ and $H_{\omega_2}$ be in bi-interpretation synonymy?

This question concerns the possibility of the bi-interpretation synonymy of the structure
$\langle
H_{\omega_1},\in\rangle$, consisting of the hereditarily countable sets, and the structure $\langle ...

6
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1
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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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### The Parovichenko cardinal, is it equal to $\max\{\aleph_2,\mathfrak p\}$?

Let us define the Parovichenko cardinal $\mathfrak{P}$ as the largest cardinal $\kappa$ such that each compact Hausdorff space $K$ of weight $w(K)<\kappa$ is the continuous image of the remainder $...

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### Is it still an open problem whether $\mathbb{R}^\omega$ is normal in the box topology?

On page 205 of his Topology textbook, James Munkres made an interesting remark:
It is not known whether $\mathbb{R}^\omega$ is normal in the box topology. Mary-Ellen Rudin has shown that the answer ...

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1
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### A new cardinal characteristic of the continuum?

Let $\kappa$ be the smallest cardinality of a family $\mathcal F$ of subsets of $\omega$ such that for any bijective function $f:A\to B$ between disjoint infinite subsets of $\omega$ there exists a ...

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### Mathematical Evidence Backing $|\mathbb{R}|=\aleph_2$

The "true" size of the real line, $\mathbb{R}$, has been the subject of Hilbert's first problem. Due to the Goedel and Cohen's work on the inner and outer models of $\text{ZFC}$, it turned ...

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### Strong Total Failures vs. Weak Instances of the Generalized Continuum Hypothesis

The exponentiation operator inflicts a subtle information loss on the transfinite numerical equations, pretty similar to the case of $a^2=b^2 \nRightarrow a=b$ in real numbers. In fact, for the ...

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### On the probability of the truth of the continuum hypothesis

First note that there exists a natural measure $\mu$ on $P(\omega \times \omega)$, inherited from the Lebesgue measure on the reals (by identifying the reals with $P(\omega)$ and $\omega$ with $\omega ...

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3
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### Example of an $\omega_1$ decreasing chain of dense semicontinua?

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 < \...

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1
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### 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 (...

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### 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 ...

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### 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 ...

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2
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### 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 $\beth$ ...

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### Does $V = \textit{Ultimate }L$ imply GCH?

In his Midrasha Mathematicae lectures ("In Search of Ultimate $L$", BSL 23 [2017]: 1–109), Woodin notes that $V = \textit{Ultimate }L$ implies $\textrm{CH}$ (Theorem 7.26, p.103). Is it known whether $...

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### Is this Variation of the Continuum Hypothesis Inconsistent with ZFC or ZF?

It is a well-known fact that the Generalized Continuum Hypothesis is undecidable from ZFC. For similar sentences $\phi$, this is simply equivalent to ZFC having a model $M$ for which $M\models\phi$.
...

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### Is it known whether or not $\aleph_\alpha=\beth_\alpha$ can be proven by ZFC?

Given a cardinal number $\aleph_\alpha$, is it known whether or not $\aleph_\alpha=\beth_\alpha$ is independent of ZFC?
One could define $\mathrm{CH}(\aleph_\alpha)$ as $\aleph_\alpha=\beth_\alpha$. ...