Do any proofs/properties rely on the distinction between some uncountable size and a larger uncountable size, in order for the proof/property to hold?

score 213 · Answer 8 · 2020-05-14 06:28:55

Assuming the falsity of continuum hypothesis is added as an axiom, how do we produce sets with cardinality in-between $\Bbb N$ and $\Bbb R$?

score 123 · Answer 9 · 2026-03-21 17:06:52

123

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Find the cardinality of the quotient of $\mathbb R$ in respect to R

Published on 21 Mar 2026 - 17:06

score 50 · Answer 10 · 2020-05-16 18:10:21

50

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Prove or disprove statement about cardinality of union

Published on 16 May 2020 - 18:10

score 84 · Answer 11 · 2020-05-17 11:38:33

84

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Is there a way of proving that if $A$ is an infinite set and $x\not\in A$ then $|A|=|A\cup\{x\}|$ without relying on the axiom of choice.

Published on 17 May 2020 - 11:38

score 202 · Answer 12 · 2020-05-17 17:36:27

202

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Manifold that's not $\aleph_1$-separable

Published on 17 May 2020 - 17:36

score 175 · Answer 13 · 2026-03-25 07:53:26

175

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On the cardinality of the closure of a subset of a dense subset

Published on 25 Mar 2026 - 7:53

score 54 · Answer 14 · 2020-05-18 13:05:14

54

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Let $X=\left \{ f:\mathbb{N} \to \mathbb{N} \;\;\text{sucht that } f(f(x))=x \;\;\;\forall x \in \mathbb{N} \right \}$

Published on 18 May 2020 - 13:05

score 195 · Answer 15 · 2026-03-28 05:39:07

195

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Strictly increasing sequence of ordinals indexed by a regular cardinal: $\text{cf}(\bigcup\{\alpha_i:i\in\kappa\})=\kappa.$

Published on 28 Mar 2026 - 5:39

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Do any proofs/properties rely on the distinction between some uncountable size and a larger uncountable size, in order for the proof/property to hold?

Assuming the falsity of continuum hypothesis is added as an axiom, how do we produce sets with cardinality in-between $\Bbb N$ and $\Bbb R$?

Find the cardinality of the quotient of $\mathbb R$ in respect to R

Prove or disprove statement about cardinality of union

Is there a way of proving that if $A$ is an infinite set and $x\not\in A$ then $|A|=|A\cup\{x\}|$ without relying on the axiom of choice.

Manifold that's not $\aleph_1$-separable

On the cardinality of the closure of a subset of a dense subset

Let $X=\left \{ f:\mathbb{N} \to \mathbb{N} \;\;\text{sucht that } f(f(x))=x \;\;\;\forall x \in \mathbb{N} \right \}$

Strictly increasing sequence of ordinals indexed by a regular cardinal: $\text{cf}(\bigcup\{\alpha_i:i\in\kappa\})=\kappa.$

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