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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11 votes
1 answer
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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 ...
Mizar's user avatar
  • 3,096
9 votes
2 answers
1k 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 $\beth$ ...
Jon's user avatar
  • 101
19 votes
2 answers
2k views

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 $...
Beau Madison Mount's user avatar
18 votes
0 answers
820 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 $M\models\phi$. ...
Keith Millar's user avatar
  • 1,242
10 votes
1 answer
996 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 $\aleph_\alpha=\beth_\alpha$. ...
Zetapology's user avatar
2 votes
0 answers
334 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 ...
V. Sciortino's user avatar
4 votes
0 answers
203 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-...
Hanul Jeon's user avatar
  • 2,774
45 votes
1 answer
3k views

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 ...
Thomas Benjamin's user avatar
9 votes
3 answers
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 ...
Chill2Macht's user avatar
  • 2,622
3 votes
1 answer
1k 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?
Mikhail Katz's user avatar
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3 votes
1 answer
429 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{...
Morteza Azad's user avatar
7 votes
1 answer
475 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?
Ohad Drucker's user avatar
3 votes
1 answer
277 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 $\subseteq^*$. ...
Dominic van der Zypen's user avatar
19 votes
1 answer
794 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 ...
Wojowu's user avatar
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1 vote
2 answers
561 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 ...
Thomas Benjamin's user avatar
14 votes
3 answers
1k views

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 ...
Jesse Elliott's user avatar
10 votes
3 answers
2k views

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 ...
12 votes
3 answers
733 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 ...
Noah Schweber's user avatar
16 votes
1 answer
2k 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, ...
Wojowu's user avatar
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14 votes
0 answers
365 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 ...
James E. Reid's user avatar
11 votes
2 answers
1k 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 ...
user avatar
13 votes
1 answer
582 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 ...
Monroe Eskew's user avatar
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12 votes
1 answer
501 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 ...
user avatar
7 votes
3 answers
1k views

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 ...
user avatar
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 $GCH$?
user avatar
12 votes
1 answer
874 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 ...
Stefan Geschke's user avatar
5 votes
1 answer
228 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) $W\models CH,$ ...
Mohammad Golshani's user avatar
4 votes
2 answers
723 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. $\text{ZF}+\text{GCH}\vdash \text{AC}$. ...
user avatar
19 votes
4 answers
3k views

A New Continuum Hypothesis (Revised Version)

Define $N_n$ as $n$ th natural number: $N_0=0, N_1=1, N_2=2, ...$. What happens after exponentiation? We have the following equation: $2^{N_n}=N_{2^{n}}$. (Which says: For all finite cardinal $n$ ...
user avatar
5 votes
1 answer
346 views

An explicit construction of reals added after some forcing notions

Consider the forcing notion(s) introduced by Friedman (or Mitchell or Neeman) for adding a club subset of $\omega_2$ by finite conditions. In the generic extension CH fails, but I can't see the reals ...
Mohammad Golshani's user avatar
6 votes
2 answers
747 views

The First Failure of GCH in Large Cardinals Smaller than Measurables

A well known theorem by Scott says: If $\kappa$ is a measurable cardinal and $\mu$ a normal measure on it and $\mu (\lbrace\lambda\in\kappa~|~2^{\lambda}=\lambda^{+}\rbrace)=1$ then $2^{\kappa}=\...
user avatar
8 votes
0 answers
355 views

Ultrapowers of Banach spaces without the continuum hypothesis

Let $\mathcal{U}$ be a non-trivial ultrafilter on the set of integers $\mathbb{N}$, and let $C(K)$ denote the Banach space of continuous functions on a compact $K$. Under the continuum hypothesis CH, ...
M.González's user avatar
  • 4,301
15 votes
2 answers
1k views

Open coloring axiom vs. CH

Is there a simple, direct proof that the open coloring axiom contradicts CH (straight from the definitions, no machinery allowed)? The separable metric spaces version of OCA, if that helps. Edit: ...
Nik Weaver's user avatar
17 votes
2 answers
877 views

Intersection of compact sets in the unit interval

Let $\mathscr K$ be an uncountable set such that every $K\in\mathscr K$ is a compact subset of $[0,1]$ with positive Lebesgue measure. Does it then follow that there exists an uncountable $\mathscr A\...
TaQ's user avatar
  • 3,348
6 votes
1 answer
410 views

The Ground Axiom for special statements of set theory

The Ground Axiom (GA), introduced by Hamkins and Reitz, asserts that the universe is not a nontrivial set forcing extension of any inner model, and it is known that GA is consistent relative to ZFC. ...
Mohammad Golshani's user avatar
8 votes
1 answer
502 views

Making all cardinals countable and its HOD

Suppose that $G$ is $Col(\omega, <Ord)$-generic over $V$ and let $W=HOD^{V[G]}$. Is $CH$ true in $W$? In general, what can we say about the behaviour of the power function in $W$? Update. Are the ...
Mohammad Golshani's user avatar
20 votes
2 answers
1k views

Why does CH imply that there is a unique ultrapower of $\mathbb{N}$?

I've read these words: "How many ultra products $∏_Uℕ$ exist up to isomorphism, where $U$ is a non-principal ultrafilter over $ℕ$? If continuum hypothesis(CH) holds, then obviously just one ..." i ...
bestian tang's user avatar
13 votes
3 answers
1k views

When was the continuum hypothesis born?

The question Solutions to the Continuum Hypothesis states that the continuum hypothesis was posed by Cantor in 1890. In http://en.wikipedia.org/wiki/Continuum_hypothesis the year 1878 is quoted ...
user avatar
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 ...
Nick Worrall's user avatar
5 votes
2 answers
2k views

Are all models of ZF + DC + "All set of reals are lebesgue measurable" also models of CH? [duplicate]

Possible Duplicate: Lebesgue Measurability and Weak CH I have studied a little set theory and I found that Solovay constructed a model of ZF+DC+"All set of reals are Lebesgue measurable" and I ...
Iván Ongay Valverde's user avatar
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 ...
user avatar
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=...
John Bentin's user avatar
  • 2,427
1 vote
1 answer
568 views

Cardinal Arithmetic, foundations and constructive math

This is not my area but a question occurred to me that I can not find the answer to. There is a very strong axiom of constructibility which ironically gives us highly non-constructive math (GCH is one ...
Najdorf's user avatar
  • 731
21 votes
1 answer
2k views

The Continuum Hypothesis and Countable Unions

I recently edited an answer of mine on math.SE which discussed the implication of the two assertions: $AH(0)$ which is $2^{\aleph_0}=\aleph_1$, and $CH$ which says that if $A\subseteq 2^{\omega}$ and ...
Asaf Karagila's user avatar
  • 38.1k
8 votes
2 answers
867 views

Iterated forcing and CH

I need some help with this theorem: if $P_\beta=\langle P_\alpha,\dot{Q}_\alpha:\alpha\leq\beta\rangle$, $\beta<\omega_2$, is a CSI of proper forcings, $P_\alpha\Vdash \lvert \dot{Q}_\alpha\rvert\...
Ruetta's user avatar
  • 81
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 ...
Gil Kalai's user avatar
  • 24.2k
26 votes
4 answers
3k views

When $2^\alpha = 2^\beta$ implies $\alpha=\beta$ ($\alpha,\beta$ cardinals)

Sorry if this is a silly question. I was wondering, under what axioms of set theory is it true that if $\alpha$,$\beta$ are cardinals, and $2^\alpha=2^\beta$, then $\alpha=\beta$? Do people use these ...
Steve D's user avatar
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