# 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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### Validity of certain arguments about the countability of infinite sets

I am trying to get an understanding, in layman's terms / on an intuitive level, why some arguments about the countability of infinite sets are valid, and some arguments which seem almost identical on ...
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### Showing a cardinal is regular

I've been thinking about this question, but to no avail and I've got to ask. How to show that for $\kappa\geq\aleph_0,$ $\mu=\min\{\lambda: \kappa^{\lambda} > \kappa\}$ is regular? If I wanted a ...
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### Another characterization of the cofinality?

Is it true that $cf(\kappa)=\min \{\lambda:\ \kappa^{\lambda}>\kappa\}$? $cf(\kappa)$ is certainly $\geq$ than that minimum since $\kappa^{cf(\kappa)}>\kappa$, but I don't know how to tackle ...
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### What are some applications of large cardinals?

Most mathematicians don't often encounter cardinalities larger than that of the continuum, it seems? What are some results outside of pure set theory/logic that rely on the properties of larger ...
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### Equivalence Relations and Cardinality

I'm looking at the question below from a past paper: What is an equivalence relation? Say that two sets $X$ and $Y$ are related via the relation $\rho$ if $X$ and $Y$ have the same cardinality. Prove ...
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### Without the Axiom of Choice, $\aleph_0<2^{\aleph_0}$ implies $\aleph_1\le 2^{\aleph_0}$?

Question: In ZF (so AC does not necessarily hold) does the following claim hold? $\aleph_0<2^{\aleph_0}$ implies $\aleph_1\le 2^{\aleph_0}$ This question arose to me when reading the top ...
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### Comparability of a set and a subset of power set.

It's well known that for any set $A$, $A < P(A)$. But now, I have some question that, WITHOUT AC, can we guarantee that $A \leq X$ or $X \leq A$ whenever $X \subseteq P(A)$? Thank you.
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### Equinumerousity of operations on cardinal numbers

I want to prove for all Cardinal numbers $a$, $b$, $c$ that: $(a \cdot b)^c =_c a^c \cdot b^c$ $a^{(b+c)} =_c a^b \cdot a^c$ $(a^b)^c =_c a^{b \cdot c}$ I know that for 1. it's enough to show ...
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### Question about $\aleph$-fixed point

I am working through a proof on cardinals I found and can't reason some of the steps. The proposition is that there is an $\aleph$-fixed point, i.e. there is an ordinal $\alpha$ (which is ...
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### Regarding the axiom $2^\kappa = 2^{\kappa^+}$ for regular cardinals $\kappa$, and its relationship to a couple of other axioms.

(Take ZFC as background.) The following two statements both follow from GCH: ICF. Injective continuum function. The continuum function (i.e. $\kappa \mapsto 2^\kappa)$ is injective. NJA....
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### What is $n^{\aleph_0},n\in\mathbb N$

Can I say that $$n^{\aleph_0}=2^{\aleph_0\log_2n}=2^{\aleph_0}=\aleph_1$$
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### Does $\operatorname{card}(X) < \operatorname{card}(Y)$ imply $\operatorname{card}(X^2) < \operatorname{card}(Y^2)$ without choice?

I looked to see if this question was already posted, but did not find anything. Please let me know if this is a duplicate. Assume $X, Y$ are infinite sets such that there is an injection $X \to Y$ ...
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### Number of subsets with the same cardinality

Suppose we have a set $S$ with cardinal number $n$, such that $n+n = n$. Consider the set, T, of all subsets with cardinality $n$. How can I show that the cardinality of $T$ is $2^n$? (Without the ...
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### Constructing $2^\kappa$ vectors with a certain property

Take an infinite rectangular array with $\kappa$ columns and $2^\kappa$ rows where $\kappa$ is some infinite cardinal. Can you fill it up with at most $\kappa$ different elements in such a way that ...
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### $y\mapsto 10^y$ Bijection between integers and powers of 10

Say $y$ and $x \in \mathbb N$ such that $10^y = x$ MSE research gives me this: $y\mapsto 10^y$ For some of you to see the bijection between the integers and powers of 10 will be obvious, but I ...
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### Continuum hypothesis : Number of connected components of $A^c$ in $\Bbb R$

Let $A \subset \Bbb R$ with a countably infinite number of connected component (for the usual topology). What can be the cardinal of the set of the connected components of $A^c$? With continuum ...
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### Finding the cardinality of a collection of lines?

I'm trying to find the cardinality of the set of lines such that the lines are not vertical, the slopes are a natural number, and the line has a natural number for a $y$-intercept. Because there are ...
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### Showing the cardinality of a bounded shape in the xy plane?

I am trying to show that the cardinality of the space between $x^2+y^2<1$ and $x+y>1$ is the same as the cardinality of real numbers I haven't a clue where do begin with something like this. I ...
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### Having trouble understanding aspects of cardinality.

I am having trouble understanding the meaning of $\omega_\alpha$, I thought it simply meant that it was the initial $\alpha$ segment of $\omega$, but then that wouldnt make sense if $\aleph_0$ is ...
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### Possible Cardinalities of the union of two sets

So the question is: What are the possible cardinalities of the union of the two sets $A$ (where $[A] = 5$) and $B$ (where $[B] = 9$) So, the smallest $[A \cup B]$ is when all elements of $A$ ...
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### $\infty$ and $-\infty$ are to $\aleph_0$ / $\beth_0$ as “what” is to $\beth_1$?

So, I recently asked a question about whether $\beth_1$ had a negative, and I was promptly reprimanded because I confused $\aleph_0$ with $\infty$. Therefore, to help me understand the concepts ...
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### Is noetherianity really about cardinals or ordinals?

If $\kappa$ is any cardinal, then one may define a "$\kappa$-Noetherian" ring as a ring such that for any module that has a generating set $S$ satisfying $|S|< \kappa$, then any submodule also has ...
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### The weight of $X$ is $\aleph_0$ iff $X$ is second countable

Let $\omega(X)=\min\{|\mathcal{B}|:\mathcal{B} \mbox{ is a base of the topology of } X\}$. I want to make sure whether the following statement is true : $\omega(X)=\aleph_0$ iff $X$ is second ...
Why does the tree property hold for regular cardinals but not singular cardinals? (I.e. There exists a tree of height $\kappa$ with countable levels and no cofinal branch for $\kappa$ a singular ...