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.
116 questions
18 votes
3 answers
747 views
Possible cardinalities of maximal chains in ${\cal P}(\omega)$
Let ${\cal P}(\omega)$ denote the power-set of $\omega$. We order it by set inclusion $\subseteq$ and say that ${\cal C}\subseteq {\cal P}(\omega)$ is a chain if for all $A, B\in {\cal C}$ we have $A\...
- 53.4k
6 votes
1 answer
308 views
Erdős–Sierpiński duality for $s^0$
A set of reals $A$ is Marczewski null, written $A \in s^0$, if, for any nonempty perfect $P$, there is a nonempty perfect $P' \subseteq P$ so that $P' \cap A = \emptyset$. This is quite different from ...
- 2,063
3 votes
1 answer
248 views
Non-measurable sum of Borel measurable functions into a Banach space
Assume that the axiom of choice and continuum hypothesis hold. I will call a discrete probability space $(X,2^X,\mu)$ diffused if $\mu(\{x\})=0$ for each $x\in X$. Different authors give different ...
- 103
2 votes
0 answers
211 views
Fragments of set theory required to prove the independence of CH
What are the smallest fragments of set theory known to be sufficient to prove Cohen's independence theorems that if ZF is consistent then so is ZF plus the negation of the continuum hypothesis CH, or ...
- 5,493
31 votes
1 answer
2k views
Is there an “opposite” hypothesis to the (Generalized) Continuum Hypothesis?
There are many questions on this site about the (Generalized) Continuum Hypothesis, its philosophical or epistemological justifications, and various attempts at “solving” it. Because one such ...
- 38k
10 votes
0 answers
265 views
Does Truss theorem on powerset intervals restricted to ordinals, entail the axiom of choice?
This comment says: An old theorem of Truss says that if there is some $\alpha$ such that there are no chains of type $\alpha$ (of distinct cardinals) between $X$ and $\mathcal P(X)$, then the axiom ...
- 13k
11 votes
2 answers
2k views
Why do some mathematicians think Gödel and Cohen have not closed the Continuum Hypothesis?
Gödel, in 1938, showed that CH is consistent with ZFC. In 1963, Paul Cohen proved the opposite: CH can be false in some models of ZFC. Together, these results mean that within ZFC alone, CH can’t be ...
- 3,296
9 votes
0 answers
237 views
Projective ordinals in ZF
$\newcommand{\proj}[1]{\boldsymbol{\delta}^1_{#1}}$ Let $\proj{n}$ be the supremum of ordinals $\alpha$ so that there exists a subjection $f: \mathbb{R} \to \alpha$ with $\{(x, y) \in \mathbb{R}^2: f(...
- 2,063
5 votes
1 answer
443 views
Minimal cardinal for families of sequences without a common lower bound
Consider a collection $A$ of positive sequences $x=(x_n)_{n\geq0}$. It is easy to see¹ that if $A$ is countable, then there exists a positive sequence $y$ such that $y=O(x)$ for every $x\in A$ (...
- 4,131
1 vote
0 answers
194 views
Does a bijection between well orders of two sets imply a bijection between the sets? [closed]
We know that whether $|P(x)|=|P(y)|$ implies $|X|=|Y|$ is dependent on CH. Let $W(X)$ be the set of all well orders over $X$. Does $|W(X)|=|W(Y)|$ imply $|X|=|Y|$? Is the answer dependent on CH? More ...
- 77
5 votes
1 answer
1k views
How many well-orders of reals are there?
It's commonly known that the cardinality of the set of all well-orders on $\aleph_0$ is the continuum (correct me if I'm wrong plz). What about that of all well-orders on $\mathbb{R}$? Is there a ...
- 77
-2 votes
1 answer
226 views
Questions on continuum hypothesis [closed]
We have the notorious continuum hypothesis (CH). According to Wikipedia it states "There is no set whose cardinality is strictly between that of the integers and the real numbers." Gödel ...
- 153
16 votes
2 answers
1k views
CH in non-set theoretic foundations
I asked this question on stack exchange and got little attention, barring a nice example I intend to look into. The original post can be found here: https://math.stackexchange.com/q/4941233/1053681 I ...
- 261
2 votes
1 answer
295 views
Freiling's axiom of symmetry and CH - need some help
Reading this there is a proof with (1) Suppose $2^\kappa = \kappa^+$. Then there exists a bijection $\sigma : \kappa^+ \to \mathcal{P}(\kappa)$. Setting $f : \mathcal{P}(\kappa) \to \mathcal{P}(\...
- 37
20 votes
5 answers
2k views
Uniqueness results that follow from CH
Recently, Joel David Hamkins presented a historical thought experiment that shows that CH could have been adopted as an axiom if we had been using the hyperreal field $\mathbb{R}^*$ instead of $\...
17 votes
2 answers
2k views
Is it consistent with ZFC that the real line is approachable by sets with no accumulation points?
Let $P$ denote the following proposition: There exists a set $S$ of subsets of $\mathbb{R}$ such that $S$ is totally ordered by inclusion; each member of $S$ has no accumulation points; the union of ...
- 902
3 votes
1 answer
374 views
Is each of the infinite statements of the Generalized Continuum Hypothesis independent?
What I know so far is that: The Continuum Hypothesis (CH) states $\nexists\mathbb{S}|\beth_0<|S|<\beth_1$ $\beth_0$ being equal to $\aleph_0$, and $\beth_n$ being equal to $2^{\beth_{n-1}}$ ...
- 147
3 votes
0 answers
508 views
Can the Collatz conjecture be independent of ZFC? [closed]
It is known that the Continuum Hypothesis is independent of ZFC. The formulation of the Collatz conjecture looks somehow more simple than that of the Continuum Hypothesis. Is it possible that the ...
- 708
4 votes
1 answer
599 views
How to settle the Generalized Continuum Hypothesis when there are urelements?
Work in $\sf ZFCA$ and permutation models has preceded forcing by several decades. Was it used to settle the question of the Generalized Continuum Hypothesis $\sf GCH$ when urelements are admitted? I ...
- 13k
8 votes
3 answers
753 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 ...
- 902
47 votes
10 answers
6k 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
95 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 ...
- 7,128
1 vote
1 answer
164 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 ...
- 231
5 votes
1 answer
306 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 ...
- 505
13 votes
1 answer
714 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 \...
- 5,120
12 votes
1 answer
377 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\...
- 53.4k
3 votes
0 answers
136 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 ...
- 31
12 votes
2 answers
1k 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,...
- 33.5k
4 votes
1 answer
1k 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: ...
- 481
7 votes
1 answer
505 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 ...
- 71
3 votes
1 answer
201 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{ ...
- 2,308
2 votes
0 answers
141 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 ...
- 2,308
1 vote
0 answers
109 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}^{\...
- 141
1 vote
1 answer
248 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, ...
- 13k
2 votes
0 answers
188 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) } ...
- 13k
-2 votes
1 answer
207 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 \, (...
- 13k
4 votes
1 answer
412 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, ...
user173426
22 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,...
18 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 ...
- 88.1k
2 votes
1 answer
247 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 ...
- 13k
1 vote
0 answers
212 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 ...
- 13k
1 vote
1 answer
201 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 ...
- 13k
11 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 $f$ ...
- 13k
16 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 ...
- 385
2 votes
1 answer
318 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 ...
- 1,425
11 votes
1 answer
1k 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? ...
- 2,442
3 votes
0 answers
406 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 ...
- 2,625
-3 votes
2 answers
303 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 ...
- 10.4k
4 votes
0 answers
262 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 $\...
- 33.5k
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 ...
- 263