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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
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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 ...
Julian Newman's user avatar
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 ...
Christopher King's user avatar
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 ...
Y.Z.'s user avatar
  • 231
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 ...
Matteo Casarosa's user avatar
13 votes
1 answer
615 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 \...
David Gao's user avatar
  • 1,297
11 votes
1 answer
279 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\...
Dominic van der Zypen's user avatar
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 ...
GAW's user avatar
  • 31
12 votes
2 answers
878 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,...
Mohammad Golshani's user avatar
4 votes
1 answer
816 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: ...
M. Solomon's user avatar
7 votes
1 answer
302 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 ...
Pavlos S's user avatar
3 votes
1 answer
163 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{ ...
Lorenzo's user avatar
  • 2,174
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 ...
Lorenzo's user avatar
  • 2,174
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}^{\...
T.Ch.'s user avatar
  • 131
1 vote
1 answer
174 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, ...
Zuhair Al-Johar's user avatar
2 votes
0 answers
158 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) }...
Zuhair Al-Johar's user avatar
-2 votes
1 answer
169 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 \, (...
Zuhair Al-Johar's user avatar
4 votes
1 answer
341 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, ...
user avatar
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 ...
Timothy Chow's user avatar
  • 79.3k
2 votes
1 answer
170 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 ...
Zuhair Al-Johar's user avatar
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 ...
Zuhair Al-Johar's user avatar
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 ...
Zuhair Al-Johar's user avatar
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 $f$ ...
Zuhair Al-Johar's user avatar
15 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 ...
Jiu's user avatar
  • 375
2 votes
1 answer
219 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 ...
Ember Edison's user avatar
10 votes
1 answer
684 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? ...
Guozhen Shen's user avatar
  • 1,288
3 votes
0 answers
326 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 ...
Mozibur Ullah's user avatar
-3 votes
2 answers
237 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 ...
Anixx's user avatar
  • 9,496
4 votes
0 answers
197 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 $\...
Mohammad Golshani's user avatar
6 votes
2 answers
1k 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 ...
Rivers McForge's user avatar
8 votes
0 answers
255 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 ...
zeb's user avatar
  • 8,623
0 votes
0 answers
135 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 \...
Zuhair Al-Johar's user avatar
9 votes
0 answers
253 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 ...
Todd Eisworth's user avatar
9 votes
0 answers
374 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\...
Todd Eisworth's user avatar
11 votes
0 answers
459 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"...
Todd Eisworth's user avatar
6 votes
1 answer
589 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 ...
Joel David Hamkins's user avatar
6 votes
1 answer
650 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 ...
user21820's user avatar
  • 2,731
7 votes
1 answer
225 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 ...
Damian Sobota's user avatar
11 votes
1 answer
386 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 $...
Taras Banakh's user avatar
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 ...
Keshav Srinivasan's user avatar
7 votes
1 answer
428 views

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 ...
Taras Banakh's user avatar
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 ...
Morteza Azad's user avatar
2 votes
1 answer
376 views

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 ...
Morteza Azad's user avatar
29 votes
2 answers
2k views

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 ...
Mohammad Golshani's user avatar
2 votes
3 answers
233 views

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 < \...
Daron's user avatar
  • 1,761
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 (...
Morteza Azad's user avatar
14 votes
1 answer
597 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 ...
user avatar
7 votes
0 answers
273 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 ...
David Feldman's user avatar
11 votes
1 answer
491 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 ...
Mizar's user avatar
  • 3,106