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.
117 questions
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Consistency of a measure witnessing a strengthening of Freiling’s axiom of symmetry
I'm interested in Freiling's axiom of symmetry and I specifically wonder if it may be proven from more basic axioms about measures on $\mathbb R^n$, in the sense that there is a sequence of measures $\...
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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\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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(...
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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 $x\in A$ (...
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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 ...
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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 ...
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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 ...
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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 ...
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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}(\...
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