# 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.

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### Prove that for $S$ a maximal mean less subset, $|S|=\aleph$.

We say that $S \subset \mathbb{R}$ is a mean less subset if for all $a \in S, b \in S$ we get that $(a+b)/2 \notin S$. Prove that for $S$ a maximal mean less subset, $|S|=\aleph$. Clearly we know ...
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### Stuck with a proof regarding cardinality

Problem: For any set $A$, finite or infinite, let $B^{A}$ be the set of all functions mapping $A$ into the set $B=\{0,1\}$. Show that the cardinality of $B^{A}$ is the same as the cardinality of the ...
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### Let $f : N → Y$ be a map having right inverse $g$. Prove that $Y$ is at most countable? [duplicate]

I know this implies $F$ is surjective, but not sure if this helps.
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### How can we show that the set of all polynomials with integer coefficients denumerable?

Is the set of all polynomials with integer coefficients, denumerable? I know that this set is countable but how can I show that it is denumerable as well.
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### Proof that $\mathbb{R}$ is not countable

I know that this proof may sound ridiculous, but I'm really curious to find out if it's logically correct(and whether there are some circularities). Since $|[0,1]|=|\mathbb{R}|$, we have to simply ...
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### Intuition of countable sets using discrete sets

I'm trying to think of a way to intuitively explain myself what a countable set means. So far as I understand it, in very simple words, a set $S$ is countable iff you can "name" or "...
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### Ordinals and Cardinals larger than the fixed points of $α↦ω_α$ and $α↦\aleph_α$?
Surely there is no limit to how high we can go, so how do we talk about ordinals and cardinals higher than the fixed points of the functions $α↦ω_α$ and $α↦\aleph_α$? Is the power set of the $\aleph$-...