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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some elementary questions about cardinality

I know little about set theory and while reading some Algebra proof I had difficulty on some details. So my questions are : If $X$ is an infinite set and $Y$ is the set of all finite subsets of $X$, ...
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2answers
153 views

What is the value of $\beth_{\omega_1}^{\aleph_0}$?

It is well known that $\beth_\omega^{\aleph_0} = \beth_{\omega+1}$. This follows since for strong limit $\kappa$, we have $\kappa^\kappa = \kappa^{\mathrm{cf}(\kappa)}.$ Question. To the extent that ...
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261 views

Uncountable Cardinals without AC

I am doing an exercise, proving that without AC or Replacement that there are uncountable cardinals. As a point of reference I looked at the proof in Kunen's "The Foundations of Mathematics" that ...
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5answers
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$\mathrm{card} ( \mathbb{Q})=\mathrm{card}( \mathbb{Q^c})$: Overcoming Wrong Intuition

This is a widespread intuitive argument, asserting that $\mathrm{card} ( \mathbb{Q})=\mathrm{card}( \mathbb{Q^c})$: Between any two rational numbers there's an irrational one and vice versa. So ...
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2answers
105 views

Question about proof of Hessenberg: $\kappa \cdot \lambda = \lambda$

The following is a theorem: (Hessenberg) Let $1 \le \kappa \le \lambda$ where $\lambda$ is an infinite cardinal. Then $\kappa \cdot \lambda = \lambda$. The proof in the book proceeds by transfinite ...
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1answer
87 views

Cardinal Arithmetic: $\beth_\omega$

Is the following equality independent of ZFC: $$\bigcup_{\aleph_{\beta}<\beth_{\omega}}2^{\aleph_{\beta}}=\beth_\omega $$ Consistent: Now that the equality is consistent with ZFC since it holds ...
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95 views

Landing between $\beth_\lambda$ and $\beth_{\lambda+1}.$

Main Question. Is it consistent with ZFC that there exists a limit ordinal $\lambda$ and a cardinal number $\nu$ satisfying $$\beth_\lambda < \beth_\lambda^\nu < \beth_{\lambda+1}?$$ I am also ...
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121 views

Existence of a regular uncountable $\aleph_{\alpha}$ without $\mathsf{AC}$

Set theory (Jech) $\text{p.}\;27:$ It is an open problem whether one can prove without the axiom of choice that there exists a regular uncountable $\aleph_{\alpha}\;($the informed guess is that ...
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1answer
166 views

Injective or constant function on measurable cardinal

I'm working on the following question: Let $\kappa$ regular cardinal, and $f:\kappa \rightarrow \kappa$. Prove that $\{\alpha < \kappa\mid f|_\alpha : \alpha \rightarrow \alpha\}$ is a ...
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1answer
157 views

Aleph arithmetic question

We want to prove that: $$\aleph_2^{\aleph_0} = \aleph_2\aleph_1^{\aleph_0}$$ My idea was to approach this by doing a Schroder-Bernstein style argument and proving this by showing two inequalities, ...
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126 views

Reference request, self study

I'm looking for references (books/lecture notes) for : Cardinality without choice, Scott's trick; Cardinal arithmetic without choice. Any suggestions ? Thanks in advance.
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Cardinality of a set A is strictly less than the cardinality of the power set of A

I am trying to prove the following statement but have trouble comprehending/going forward with some parts! Here is the statement: If $A$ is any set, then $|A|$ $<$ $|P(A)|$ Here is what I ...
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2k views

An infinite subset of a countable set is countable

In my book, it proves that an infinite subset of a coutnable set is countable. But not all the details are filled in, and I've tried to fill in all the details below. Could someone tell me if what I ...
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2answers
139 views

Constructing a vector space of dimension $\beth_\omega$

I'm trying to solve Exercise I.13.34 of Kunen's Set Theory, which goes as follows (paraphrased): Let $F$ be a field with $|F| < \beth_\omega$, and $W_0$ a vector space over $F$ with $\aleph_0 ...
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1answer
88 views

$\kappa\psi (x,X)\leq \psi (x,X)$

The $\kappa$-pseudocharacter $\kappa\psi (x,X)$ of a space $X$ at a point $x\in X$ is the smallest infinite cardinal number $\tau$ such that there exists a family $\gamma$ of $\kappa$-sets in $X$ ...
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1answer
124 views

What indexes do the subgroups of $\mathrm{GL}_n(\Bbb C)$ have?

Let $B_n\subset\mathrm{GL}_n(\Bbb C)$ be the group of invertible upper-triangular matrices. What is the index $[\mathrm{GL}_n(\Bbb C):B_n]?$ (By index I mean the cardinality of a coset space.) In ...
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112 views

Weak cardinal powers and singular cardinals

Suppose $\kappa > \operatorname{cf}(\kappa)$. Show that: i) if $\kappa$ strong limit then $\kappa^{<\kappa} = \kappa^{\operatorname{cf}(\kappa)}$ ii) if $\kappa$ not strong limit then ...
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1answer
673 views

How can one rigorously determine the cardinality of an infinite dimensional vector space?

Suppose $V$ is a vector space over a scalar field $F$. If $\dim(V)=n$, then $|V|=|F|^n$. How can I rigorously determine the cardinality of $V$ when $V$ is infinite dimensional? My thought was that if ...
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313 views

Cardinality of sets of functions with well-ordered domain and codomain

I would like to determine the cardinality of the sets specified bellow. Nevertheless, I don't know how to approach or how to start such a proof. Any help will be appreciated. If $X$ and $Y$ are ...
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1answer
259 views

Does any uncountable set contain two disjoint uncountable sets?

Is it true that for any uncountable $S$, there exits two uncountable subsets $S_1,S_2 \subseteq S$ with $S_1 \cap S_2 = \emptyset$? I can find no counter example, but no proof either. I am aware of ...
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139 views

Partial order on cardinalities without the axiom of choice

Cardinality can still be defined without choice, e.g. as equivalence class of equipotent sets, see Defining cardinality in the absence of choice. Injections define partial order on cardinalities by ...
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1answer
87 views

How to show that $\mathfrak s \leq \mathfrak d$

I am trying to understand why $\mathfrak s \leq \mathfrak d$. Can anyone state a proof of it? I have a proof , which I don't understand yet. My question regarding that proof is here below: At the ...
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1answer
89 views

A verification for a proof that $\omega_1 \leq \mathfrak s$

I am trying to prove that $\omega_1 \leq \mathfrak s$ where $\mathfrak s$ is the splitting number which is the smallest cardinality of any splitting family. This statement was left as an ...
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1answer
184 views

Jech's proof of Silver's Theorem on SCH

Jech's textbook proves Silver's Theorem–that if the Singular Cardinals Hypothesis holds for all singular cardinals of countable cofinality, then it holds for all singular cardinals–by breaking up the ...
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1answer
125 views

Differences in worlds with and without $\aleph_0<|S|<2^{\aleph_0}$

Paul Cohen told us that whether or not there is $S$ with \begin{equation} \aleph_0<|S|<2^{\aleph_0} \end{equation} cannot be decided within ZFC, and hence it is reasonable to work in two ...
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177 views

Orders of subgroups of Infinite Profinite Groups

This question is in some sense equivalent to my question here. A proof would answer that question in the case when the base field is perfect. Let $G$ be a profinite group of cardinality $\kappa$, ...
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2answers
215 views

Cardinal arithmetic gone wrong?

I am trying to calculate $\kappa^\lambda = \aleph_{\omega_1}^{\aleph_0}$. I know that if $\kappa$ is a limit cardinal and $0 < \lambda < \mathrm{cf}(\kappa)$ then $\kappa^{\lambda} = ...
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1answer
92 views

Why $\kappa^{<\kappa}=2^{<\kappa}$, if $\kappa$ is a regular and limit cardinal?

On Page 60, Set Theory Jech(2006) (Show that)if $\kappa$ is regular and limit, then $\kappa^{<\kappa}=2^{<\kappa}$. It's not difficult to show that $\kappa^{<\kappa}\geq2^{<\kappa}$. ...
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1answer
205 views

Why is ${\aleph_\omega}^{\aleph_1} = {\aleph_\omega}^{\aleph_0} \cdot {2}^{\aleph_1}$? [duplicate]

I am supposed to prove that ${\aleph_\omega}^{\aleph_1} = {\aleph_\omega}^{\aleph_0} \cdot {2}^{\aleph_1}$ , but I really have no idea how to start or what to do. I thought I could use the following ...
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1answer
44 views

How to prove $cf(\aleph_{\omega_1})=\omega_1$?

I am reading Jech's & Hrbacek's book "Introduction to set theory" and this is question 2.2 from chaper 9: Prove $cf(\aleph_{\omega_1})=\omega_1$ Isn't it clear from the fact that ...
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1answer
61 views

What is the cardinality of an element of an free ultrafilter?

Let $U$ be a free ultrafilter on a set $X$. I want to prove that the cardinality of every element $u\in U$ is equipotent to $X$. Is that true? Or does it lack some hypothesis?
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1answer
93 views

Why is this cardinal regular?

I have the following problem in front of me. Show that if $\kappa$ is the least cardinal such that $2^\kappa>2^{\aleph_0},$ then $\kappa$ is regular. I've scribbled this: Suppose ...
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1answer
229 views

In ZF, does there exist an ordinal of provably uncountable cofinality?

Question is in the title. In ZFC, one can prove that $\aleph_{\alpha+1}$ is regular, so there is a large source of cardinals with uncountable cofinality, but in ZF, it is consistent that ${\rm ...
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1answer
122 views

A question regarding the Power set

In the proofs that I have seen so far for showing that the power set $2^X$ of a set $X$ cannot be in bijection to $X$, the common idea is to assume that there exists a surjection $f \colon X \to 2^X$ ...
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300 views

How does the supremum of the cardinalities of a set of ordinals relate to the supremum of the ordinals?

Suppose we have a set of ordinals such that their supremum is a cardinal (not in the original set). $\sup\{\alpha\}=\kappa$ I'm interested in the supremum of the cardinalities of those ordinals: ...
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1answer
370 views

Is $2^\infty$ uncountable and is cardinality a continuous function?

I apologize if the title seems too vague, but this is how I was asked the question. So one of my friends intended to write an infinite sum like $\displaystyle \sum_{i=1}^{\infty} a_{2^i}$ . However, ...
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369 views

A cardinal exponentiation question

Suppose $\kappa$ and $\lambda$ are infinite cardinals and that $\lambda$ is regular. Kunen states somewhere that this means that we have ...
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1answer
135 views

Filter completeness question

I have a question about filters which I suspect has a very simple answer (hence my asking it here as opposed to MO): Let $F$ be a filter on an infinite set $X$. Then $F$ is "countably closed" if for ...
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2answers
52 views

How can I prove that the cadinality of a set minus a finite number of elements of it is still the same as the original set?

A is a finite subset of S, which is an infinite set. How can I prove that $|S| = |S \setminus A|$? I just finished proving that $|T \cup S|$ where $T$ is infinite and $S$ is countable is $|T|$. They ...
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1answer
111 views

Do we need choice to prove that $|\mathbb{N} \times A| = |A|$ for all infinite sets $A$?

I can't think of any way to prove it without choice.
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209 views

Dimension of $\operatorname{Hom}(V, W)$

What is the dimension of $\operatorname{Hom}(V, W)$ if at least one of the two vector spaces $V, W$ is infinite dimensional? In the sense of cardinal numbers. Thanks
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2answers
105 views

Product of a family of spaces of countable tightness

I recently learned the concept of cardinal functions and some of the definitions and theorems are not clear to me. How can we prove this theorem? Finite family of compact spaces of countable ...
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1answer
142 views

One problem about complemented subspace

Question: For every Banach space $X$ and its subspace $Y$, is there a complemented subspace $Z$ in $X$ such that $Y \subset Z \subset X $ and $\operatorname{card}(Y)=\operatorname{card}(Z)$ i.e., $Y$ ...
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1answer
222 views

Cardinality of Reals and Turing Machines

I'm a math hobbyist, so forgive me if what I ask is silly. I just learned that the cardinality of Reals is greater than the Naturals. So, because of that, there can be no turing machines which ...
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1answer
443 views

Do sets whose power sets have the same cardinality, have the same cardinality? [duplicate]

Possible Duplicate: power set cardinal equality Let $X$ and $Y$ be sets, and suppose that $|\mathscr{P}(X)| = |\mathscr{P}(Y)|$ (where $\mathscr{P}$ denotes the power set). Does it follow ...
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1answer
101 views

A club-guessing exercise

I came across this club-guessing exercise on Cardinal Arithmetic by Abraham and Magidor in the Handbook of Set Theory. Let $\kappa, \lambda$ be regular cardinals $\kappa^{++}<\lambda$ and let ...
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1answer
68 views

Cardinality of order topology?

Just out of interest: The cardinality of the Euclidean topology on the real line is $c$. In general, if $X$ is totally ordered of cardinality $\alpha$, the order topology on $X$ must have cardinality ...
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70 views

For cardinals, if $\mathfrak{a}\ne\mathfrak{b}$ then $2^\mathfrak{a}\ne 2^\mathfrak{b}$

In the usual ZF (or ZFC) set theory, let $\mathfrak{a}$ and $\mathfrak{b}$ be cardinal numbers. Is it correct that one can neither prove nor disprove the statement: $$\mathfrak{a}\ne\mathfrak{b} ...
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Is there a simple example of a ring that satifies the DCC on two-sided ideals, but doesn't satisfy the ACC on two-sided ideals?

It follows from the Hopkins–Levitzki theorem that if a ring satisfies the DCC on left ideals, then it also satisfies the ACC on left ideals. I've been trying to find a counterexample to the following ...
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Are there fields and ordered fields of every infinite cardinality?

On the top of my head, I cannot think of any fields of cardinality more than that of the reals. (It is known that the process of algebraic closure does not increase the cardinality of an infinite ...