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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Applying power set finite times

Is every infinite set $A$ smaller than a set of the form $\mathcal P (\mathcal P(\dots \mathcal P(\mathbb N)))$?
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Cardinality Proof Problem

Suppose A and B are sets such that card A ≤ card B. Prove that there exists a set C ⊆ B such that card C = card A. I know that there is only an injection from A to B. I'm having trouble showing that ...
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size of infinite strings and infinite alphabets

Please forgive the lack of formal vocabulary. Which set has a larger cardinality? A) a set of all possible countably infinite strings with a finite alphabet of symbols. B) a set of all possible ...
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Prove that $\{X \in P(Z)| X \text{ is finite}\}$ is enumerable. [duplicate]

I am not sure how to approach this problem. if you could help it would be great.
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Prove that $(0,1) =c\mathbb R$

How do I prove this knowing that $f(x) = \tan(x\pi/2)$ is a bijection between $(0,1)$ and $(0, \infty)$? We also have a bijection between $(-1,1)$ and $(0,1)$.
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Finding the cardinalty of a subset of $\mathcal{P}(\mathbb{N})$

I'm trying to find the cardinality of a certain set and I'm stuck. The problem is, we haven't learned about cardinality nor about any of its rules and equalities. We are asked to find a set whose "...
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Prove that (0,1) ⊆ R and (4,10) ⊆ R have the same cardinality [duplicate]

Hows does (0,1) and (4,10) both existing as real numbers make it have the same cardinality?
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Power Set, Theory of Linear Algebra, and Axiomatic Systems

So for homework, my History of Math professor gave us these three questions: Explain why the power set of a set S (collection of all subsets) has the same cardinality as the set [all functions from ...
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Do the adeles $\mathbb{A}$ have the same cardinality as the real numbers $\mathbb{R}$?

My question today is about the Adeles and whether they have the same cardinality as the real numbers. Certainly $\mathbb{R}$ has more elements than $\mathbb{Z}$. This is by Cantor diagonalization. ...
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definition of a $\kappa$-small simplicial set

Let $\kappa$ be a regular cardinal. A set $X$ is $\kappa$-small, if $|X|<\kappa$. What does it mean for a simplicial set $X\colon \Delta^{op}\to Sets$ to be $\kappa$-small. I can imagine at ...
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Cardinality of the Euclidean topology and the axiom of choice

It is relatively straightforward to prove that the Euclidean topology has the same cardinality as the space itself. I have sketched a proof below. The proof seems to rely quite heavily on the axiom of ...
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Trying to understand the slick proof about the dual space

In this famous MO question, a beautiful proof is given of the fact $V\cong V^\ast\iff V$ is finite dimensional. I'm trying to go through it and I'm having some trouble. First of all, I know the in ...
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Examples of uncountable fields of characteristic $p$?

Let $p$ be a prime. The axioms of a field of characteristic $p$ is definable in first order logic and form a satisfiable theory $T$. Indeed, $T$ has arbitrarily large finite models and it also has an ...
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Are cardinal numbers well defined? Or could we have something like $\aleph_{1/2}$? [duplicate]

From what I understood, cardinal numbers are defined as: $\aleph_0$ = the cardinality of $\mathbb{N}$ $\aleph_{n+1}$ = is the least cardinal number greater than $\aleph_n$ The continuum hypothesis ...
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Cardinality of a metric space

Let $X$ be an infinite set .For any two metrics $d_1,d_2$ on $X$ the identity map $i:(X,d_1)\to (X,d_2)$ is continuous. Prove that $X$ is always countable. I am not getting how to start this problem....
If a countable union of sets has card $\mathfrak{c}$, prove at least one of them has card $\mathfrak{c}$ [duplicate]
If $A=\bigcup_{n=1}^{\infty}A_n$ and $A$ has cardinality $\mathfrak{c}$, where $\mathfrak{c}$ is the cardinal of the continuum, prove that at least one of the $A_n$ has cardinality $\mathfrak{c}$.