# Tagged Questions

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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### Show that the set of powers of ten is countably infinite

My question asks: An $x \in \mathbb{N}$ is called a power of ten if there exists a $y \in \mathbb{N}$ such that $10^y = x$. Show that the set of powers of ten is countably infinite. I understand that ...
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### Impossibility of constructing a continuum-size linearly independent set in $\Bbb R$ [duplicate]

This is a response to the following exchange at Is there a quick proof as to why the vector space of $\mathbb{R}$ over $\mathbb{Q}$ is infinite-dimensional? [Bill constructs a $\aleph_0$ ...
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### Problem on infinite cardinal number

If $e$ is an infinite cardinal number and $d$ is a cardinal number satisfing $2 ≤ d ≤ 2^e$. I need to prove the following $$d^e= 2^e$$ Any help will be appreciated. Thank you in advance. .
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### Cardinality of almost everywhere continuous functions

The cardinality of continuous real functions is $|\mathbb{R}|$ but I was wondering wether allowing functions to be almost everywhere continuous would increase the cardinality or not. On the one hand, ...
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### Is it possible to define a group structure on arbitrary set? [duplicate]

Is it possible to define a group structure on arbitrary set? It is obvious for finite sets and also sets with cardinality |Q| and |R| and also we don't know that is there other cardinality betwen them ...
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### for a given $\kappa$, find a model of DLO with a greater cardinality that has a dense subset of size $\kappa$

So given a cardinal $\kappa$, i am trying to find a dense linear order with no end points of cardinality greater then $\kappa$ which has a dense subset of size $\kappa$ I think I'm almost there What ...
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### The supremum of any set of cardinals (considered as a set of ordinals) is again a cardinal.

An ordinal $\alpha$ is a cardinal iff no $\xi < \alpha$ is equivalent to $\alpha$. Now, let $A$ be any set of cardinals and $\sup(A)=\alpha$, then for $\xi < \alpha$ there is a $\beta \in A$ ...
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### How does equality $\mathfrak{m}^\mathfrak{m} = 2^\mathfrak{m}$ in $\sf{ZF}$ relate to the axiom of choice?

Usually, the equality $\mathfrak{m}^\mathfrak{m} = 2^\mathfrak{m}$ for infinite cardinal $\mathfrak{m}$ is proved like that:  2^\mathfrak{m} \leqslant \mathfrak{m}^\mathfrak{m} \leqslant (2^\...
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### Since $[0,1]$ and $\mathbb{R}$ are not homeomorphic, does that mean the cardinality of $[0,1]$ and $\mathbb{R}$ are different?

Given $[0,1]$ a closed interval on $\mathbb{R}$, we know that $[0,1]$ is compact and $\mathbb{R}$ is not, so these two spaces are not homeomorphic to each other. But homeomorphic perserves ...
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### Cardinal Addition When At Least One is Infinite

Show that if at least one of κ > 0 and λ > 0 is infinite, then κ + λ = κλ = max{κ, λ}. My proof: Assume without loss of generality, κ > λ. If λ = 1, then by definition that at least one is infinite, ...
I don't know how to prove the following: Let $K:=\lbrace G : G$ is a union of finitely many intervals with rational endpoints$\rbrace$. Prove that $K$ is countably infinite. Here is my approach:...
So, I know that $2^{\aleph_0} = \beth_1$, right? What about another number, say $10$, raised to the power of $\aleph_0$? Is $10^{\aleph_0} = \beth_1$ also true, or is $10^{\aleph_0} > \beth_1$ ...