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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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
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26 votes
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 ...
Steve D's user avatar
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32 votes
1 answer
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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
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 ...
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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
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 ...
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
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20 votes
2 answers
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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 ...
bestian tang's user avatar
19 votes
1 answer
791 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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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 $n$ ...
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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
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
14 votes
1 answer
595 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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11 votes
0 answers
453 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
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 $GCH$?
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9 votes
0 answers
371 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
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
8 votes
2 answers
865 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
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7 votes
1 answer
474 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
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
4 votes
1 answer
797 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