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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3
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0answers
271 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 ...
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2answers
170 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 ...
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172 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 $\...
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2answers
554 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 ...
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215 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 ...
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128 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 \...
7
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174 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 ...
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329 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 $A\...
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$\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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1answer
487 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
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1answer
571 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 ...
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1answer
188 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 ...
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1answer
333 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 $...
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1answer
1k 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 ...
7
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1answer
396 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 ...
28
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2answers
2k 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 ...
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1answer
341 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 ...
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2answers
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 ...
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3answers
223 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 < \...
19
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1answer
750 views

On Erdos-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 the $\neg CH$ by Erdos and Kakutani (MR0008136) as follows: Definition. ...
14
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1answer
494 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 ...
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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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1answer
420 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 ...
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742 views

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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706 views

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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1answer
734 views

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 $\aleph_\alpha=\beth_\alpha$. ...
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316 views

In Cohen's Independence of Continuum Hypothesis, can someone explain to me his definition of the $a_\delta$'s at the beginning of the paper?

Cohen's paper The independence of the Continuum Hypothesis, Proc. Natl. Acad. Sci. USA. 1963 Dec; 50(6): 1143–1148 (PMC221287) begins by invoking the Lowenheim-Skolem theorem to assert the ...
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184 views

Equivalence of Rathjen's continuum hypothesis and another form of the CH without choice

(This question is already posted on Math SE but it isn't answered, so I ask same question on this site.) The following form of a continuum hypothesis occurs in Rathjen's paper "Indefiniteness in semi-...
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1answer
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Hilbert's alleged proof of the Continuum Hypothesis in “On the Infinite”

As is known, Hilbert attempted a proof sketch of the Continuum Hypothesis in the latter part of his paper, "On the Infinite". It is also known that it is false. Has there ever been a published ...
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3answers
1k views

Does anyone understand this comment about the continuum hypothesis?

At 31:37 in his lecture titled What is a manifold? posted on Youtube, Mikhail Gromov states that if we do not allow generic functions to exist then the continuum hypothesis is "obviously" true, and ...
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1answer
742 views

Well-ordering of power set of $\omega$

Assuming initially the background set theory ZF, what is the exact status of the existence of a well-ordering on the power set of $\omega$? How much needs to be added to guarantee this?
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1answer
392 views

A Question on HOD, V and GCH

The theorem 1.1 of the following paper: Mohammad Golshani, V, HOD, and the GCH, Journal of Symbolic Logic. states that: Theorem: Assume $V\models ZFC+GCH+~\text{There exists a}~(\kappa+4)-\text{...
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1answer
405 views

Forcing the negation of CH without adding Cohen reals over L

Suppose CH + "there are no Cohen reals over L". Can we force the negation of CH without adding any Cohen real over L?
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1answer
239 views

Cardinalities of maximal towers in ${\cal P}(\omega)$

For $A,B\subseteq \omega$ we write $A \subseteq^* B$ if $A\setminus B$ is finite. We call ${\cal T}\subseteq {\cal P}(\omega)$ a tower if it is linearly quasiordered with respect to $\subseteq^*$. ...
17
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1answer
625 views

Can we find CH in the analytical hierarchy?

Recently I was talking to my friend and I have mentioned to him that it was proven that CH is not provably (over ZFC) equivalent to any statement in second-order arithmetic. However, today I found out ...
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2answers
486 views

A question regarding the relation between Freiling's Axiom of Symmetry and real-valued measurable cardnals

A major argument against Freiling's Axiom of Symmetry is the following (this from the wikipedia article of the same name): "The naive probabalistic notion used by Freiling tacitly assumes that there ...
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3answers
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Complete resolutions of GCH

Let's say that a "complete resolution of GCH" is a definable class function $F: \operatorname{Ord}\longrightarrow \operatorname{ Ord}$ such that $2^{\aleph_\alpha} = \aleph_{F(\alpha)}$ for all ...
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3answers
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Difference between ZFC and ZF+GCH

I hear that the axiom of choice (AC) derives from The generalized continuum hypothesis(GCH). And also hear that both AC and GCH are independent of Zermelo–Fraenkel set theory(ZF). So, I'm just ...
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3answers
640 views

The continuum hypothesis for packing shapes without overlapping

Consider the finite cross $C$ (=union of line segments $\overline{(0, -1)(0, 1)}$ and $\overline{(-1, 0)(1, 0)}$) and the unit half-circle $H$. It is easy to see that we may pack continuum-many ...
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1answer
1k views

How much of GCH do we need to guarantee well-ordering of continuum?

It's well known that, if GCH holds, then every cardinal can be well-ordered. However, I'm sure we don't need full power of GCH to prove it for specific cardinal, e.g. continuum. I have been wondering, ...
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322 views

Are there additive subgroups of reals of dimension 1 with no subgroups of dimension strictly between 0 and 1?

I will use $dimA$ to denote the Hausdorff dimension of a set $A \subseteq \mathbb{R}$. Being a null set means having Lebesgue measure zero. In the 1966 paper "Additive gruppen mit vorgegebener ...
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2answers
930 views

Last Status of Feferman's Conjecture on Indefinite Value of Continuum

The "true" value of $2^{\aleph_0}$ is one of the most fundamental open questions of mathematics and its philosophy. Hundreds of set theoretic results during the last century don't say anything more ...
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1answer
538 views

Does small forcing preserve CH?

Suppose CH holds and $\mathbb{P}$ is a poset of size $\omega_1$, such that forcing with $\mathbb{P}$ preserves $\omega_1$. Does forcing with $\mathbb{P}$ preserve CH? If $\mathbb{P}$ is proper then ...
12
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1answer
486 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 ...
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3answers
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PFA: A New Godel's Program & A New Large Cardinal Ladder (Updated)

We know $PFA$ implies $2^{\aleph_0}=\aleph_2$. Q1. What does $PFA$ say about other values of continuum function? Does proper forcing axiom carry any further information about values of continuum ...
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4answers
1k views

Very Large Cardinal Axioms and Continuum Hypothesis

Are very large cardinal axioms like $I_0$, $I_1$, $I_2$ consistent with $CH$ and $GCH$?
12
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1answer
742 views

Families of pairwise incomparable subsets of the integers

Certain maximal objects whose existence follows from Zorn's Lemma have received some set-theoretic attention. Examples are maximal independent families and maximal almost disjoint families. There is ...
5
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1answer
207 views

Intermediate submodels and the continuum hypothesis

Let $V$ be a model of $ZFC+GCH$ and let $V[G]$ be a generic extension of $V$ in which $CH$ fails. Question 1. Is there a model $W$ such that: 1) $V \subseteq W \subseteq V[G],$ 2) $W\models CH,$ ...
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3answers
618 views

Minimal Generalized Continuum Hypothesis & Axiom of Choice

It is well known that working in the frame of $\text{ZF}$, the Generalized Continuum Hypothesis ($\text{GCH}$) implies the Axiom of Choice ($\text{AC}$), i.e. $\text{ZF}+\text{GCH}\vdash \text{AC}$. ...