# Questions tagged [cardinals]

This tag is for questions about cardinals and related topics such as cardinal arithmetics, regular cardinals and cofinality. Do not confuse with [large-cardinals] which is a technical concept about strong axioms of infinity.

2,424 questions
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### For cardinals $a,b,b'$, if $a\ge 2$ and $b<b'$, then $a^b <a^{b'}$

I need to prove, without assuming the Axiom of Choice, that for cardinals $a,b,b'$, if $a\ge 2$ and $b<b'$, then $a^b <a^{b'}$. I have already proved that for cardinals $c,c',d,d'$, if $c\neq0$,...
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### Aronszajn tree for infinite singular cardinal [duplicate]

I've always seen Aronszajn trees being discussed on regular cardinals : Let $\kappa$ be a regular infinite cardinal. A $\kappa$-Aronszajn tree is a tree on $\kappa$ of height $\kappa$, whose ...
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### Cardinality: Injection between subsets of Uncountable set

assuming, S is infinite uncountable, I am trying to come up with injective f: (S union N) -> S. Where N is naturals. So far I created S0 which consists of infinite sequence of elements of S, such that ...
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### Cardinality of an uncountable set after union with another uncountable set, that has smaller cardinality.

When $A$ is uncountable and $B$ is a countable set, $|A\setminus B|=|A|$. How can I prove (or disprove) that $|A\setminus a|=|A|$ or $|A\cup a|=|A|$, where $A, a$ are uncountable sets and $|A|>|a|$ ...
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### Set $A$ is countably infinite if and only if there exists a bijection $f: \mathbb{N} \rightarrow A$

Using the fact that for any $A$, $A$ is countably infinite if there exists a bijection $f: A \rightarrow \mathbb{N}$, how do I prove the statement: $A$ is countably infinite if and only if $\exists$ ...
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### Prove that $|\mathbb{N}\times \mathbb{R}| = |\mathbb{R}|$

I'm trying to prove the equality between the cardinalities: $|\mathbb{N}\times \mathbb{R}| = |\mathbb{R}|$ ($\mathbb{N}\times \mathbb{R}$ is cartesian product from $\mathbb{N}$ to $\mathbb{R}$). I ...
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### linearly ordered family of sets cardinality greater than supremum of individual sets

Does there exist a set $X$ linearly ordered by $\subset$ where $|\bigcup X|>\sup\{|x|:x\in X\}$? I'm having trouble thinking around constructing one. I'm fairly certain that finite sets won't work,...
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### Is there any bijection between $\mathbb{R}$ and $\mathbb{R}^2$? [duplicate]

Is there any bijection between $\mathbb{R}$ and $\mathbb{R}^2$ ? If have then what is the mapping ? Please define the mapping. They have same cardinality then it is possible to have a bijection ...
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### Using union of countably infinite sets, I tried to prove that set of all real numbers in [0,1) is countable

Cantor's diagonal method shows that the set $S=\{x\in \Bbb R|x \in [0,1)\}$ is uncountably infinite, because there is no bijection between the set $S$ and the set of natural number $\Bbb N$. I came ...