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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8 votes
3 answers
664 views

Is "the purely probabilistic version of Freiling's axiom of symmetry" disprovable in ZFC?

I'm trying to pinpoint the "intuitive argument" for Freiling's Axiom of Symmetry. It's meant to be a "probabilistic" argument, so thinking about what seems to me to be the ...
47 votes
10 answers
5k views

What is the most "concrete-feeling" equivalent formulation of the Continuum Hypothesis that you can think of?

There are many equivalent formulations of the Continuum Hypothesis, but I think the most standard one is that there is no infinite cardinality lying strictly between the cardinality of the natural ...
2 votes
0 answers
87 views

Is there a set of ℵ₁ sequences that can dominate any sequence? [duplicate]

Is there a set $S$ of $\mathbb \aleph_1$ sequences of natural numbers such that for any sequence not in $S$, there is a sequence in $S$ that grows faster than it? Assuming the continuum hypothesis ...
1 vote
1 answer
106 views

CH and the existence of a Borel partition of small cardinality

Say $\kappa$ is small if any set of cardinality $\kappa$ has outer-Lebesgue measure zero. We know that, in the Cohen model of ZFC where CH is false, there is a Borel partition of the unit interval of ...
5 votes
1 answer
191 views

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 votes
1 answer
601 views

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 \...
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 (...
3 votes
0 answers
111 views

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 ...
11 votes
1 answer
275 views

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\...
4 votes
1 answer
797 views

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: ...
12 votes
1 answer
714 views

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,...
141 votes
12 answers
29k views

Solutions to the Continuum Hypothesis

Related MO questions: What is the general opinion on the Generalized Continuum Hypothesis? ; Completion of ZFC ; Complete resolutions of GCH How far wrong could the Continuum Hypothesis be? When was ...
7 votes
1 answer
293 views

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 ...
3 votes
1 answer
162 views

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{ ...
4 votes
1 answer
327 views

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, ...
2 votes
0 answers
107 views

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 ...
1 vote
0 answers
86 views

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}^{\...
15 votes
3 answers
2k views

Continuum Hypothesis

I am new here, so forgive me if this question does not satisfy the protocols of the site. I know there are so many equivalents to the AC (axiom of choice) and there are books that lists this ...
1 vote
1 answer
173 views

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, ...
2 votes
0 answers
157 views

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) }...
-2 votes
1 answer
168 views

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 \, (...
19 votes
3 answers
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,...
17 votes
2 answers
1k views

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 votes
1 answer
209 views

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 ...
2 votes
1 answer
168 views

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
0 answers
182 views

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 ...
10 votes
2 answers
2k views

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 $f$ ...
1 vote
1 answer
145 views

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 ...
14 votes
1 answer
2k views

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 ...
10 votes
1 answer
648 views

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? ...
5 votes
1 answer
290 views

Does strict order-preservation of powerset curtail the candidates for violation of CH?

Thus, let $\mathrm{OPP}$ be the axiom that $|A|\lt|B| \Rightarrow |2^A|\lt|2^B|$ for any sets $A$ and $B$; and, for any ordinal $\alpha$, let $\mathrm{CH}_\alpha$ be the hypothesis that $\aleph_\alpha=...
7 votes
1 answer
224 views

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 ...
3 votes
0 answers
323 views

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 ...
-3 votes
2 answers
236 views

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 ...
4 votes
0 answers
195 views

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 $\...
6 votes
2 answers
981 views

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 ...
11 votes
1 answer
653 views

Conceptual structuralism and continuum hypothesis

In Ferefman's paper 'Is the Continuum Hypothesis a definite mathematical problem?', he argues that within the philosophy of conceptual structuralism, the continuum hypothesis is not a definite ...
32 votes
1 answer
2k views

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 ...
8 votes
0 answers
252 views

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 ...
11 votes
1 answer
486 views

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 ...
0 votes
0 answers
134 views

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 \...
9 votes
0 answers
249 views

$\Sigma^2_2$ 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 ...
9 votes
0 answers
371 views

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\...
11 votes
0 answers
453 views

$\Sigma^2_1$ 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"...
6 votes
1 answer
588 views

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 votes
1 answer
637 views

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 ...
31 votes
2 answers
3k views

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 ...
12 votes
1 answer
500 views

Ground Axiom and behaviors of continuum function

The Ground Axiom ($GA$) is the assertion that the universe of sets ($V$) is not a forcing extension of any inner model $W$ by nontrivial forcing $P\in W$. Is $GA$ consistent with any possible ...
10 votes
4 answers
2k views

Very Large Cardinal Axioms and Continuum Hypothesis

Are very large cardinal axioms like $I_0$, $I_1$, $I_2$ consistent with $CH$ and $GCH$?
11 votes
1 answer
384 views

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 $...