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.
97
questions
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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 ...
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2
answers
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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 $\beth$ ...
19
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2
answers
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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 $...
18
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0
answers
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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 $M\models\phi$.
...
10
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1
answer
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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 $\aleph_\alpha=\beth_\alpha$. ...
2
votes
0
answers
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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 ...
4
votes
0
answers
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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-...
45
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1
answer
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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 ...
9
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3
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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 ...
3
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1
answer
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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?
3
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1
answer
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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{...
7
votes
1
answer
475
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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?
3
votes
1
answer
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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 $\subseteq^*$.
...
19
votes
1
answer
794
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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 ...
1
vote
2
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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 ...
14
votes
3
answers
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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 ...
10
votes
3
answers
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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 ...
12
votes
3
answers
733
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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 ...
16
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1
answer
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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, ...
14
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0
answers
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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 ...
11
votes
2
answers
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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 ...
13
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1
answer
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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
votes
1
answer
501
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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 ...
7
votes
3
answers
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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 ...
10
votes
4
answers
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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
votes
1
answer
874
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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
votes
1
answer
228
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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) $W\models CH,$
...
4
votes
2
answers
723
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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. $\text{ZF}+\text{GCH}\vdash \text{AC}$.
...
19
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4
answers
3k
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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 $n$ ...
5
votes
1
answer
346
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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 ...
6
votes
2
answers
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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}=\...
8
votes
0
answers
355
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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, ...
15
votes
2
answers
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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: ...
17
votes
2
answers
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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\...
6
votes
1
answer
410
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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. ...
8
votes
1
answer
502
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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 ...
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2
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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 ...
13
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3
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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 ...
11
votes
1
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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 ...
5
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2
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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 ...
15
votes
3
answers
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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 ...
5
votes
1
answer
290
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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=...
1
vote
1
answer
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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 ...
21
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1
answer
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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 ...
8
votes
2
answers
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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\...
141
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12
answers
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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 ...
26
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4
answers
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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 ...