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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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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209 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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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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81 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
83 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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173 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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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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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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214 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
194 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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What is known about the power set of the real number line?

Cantor's theorem states that the cardinality of a set's powerset is strictly greater than that of the set itself. This clearly applies to the reals also; if I'm not mistaken, the cardinality of the ...
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73 views

Why we use $\mathrm{Fn}(\kappa\times\lambda,2,\lambda)$ to force $2^\lambda\ge\kappa$ instead of $\mathrm{Fn}(\kappa\times \lambda,2)$

I am reading Kunen's Set Theory and I learn that, $\operatorname{Fn}(\kappa\times\omega,2)$ forces $2^{\aleph_0}\ge|\kappa|$, where $$\operatorname{Fn}(I,J,\lambda)=\{p:p\text{ ...
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1answer
58 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
87 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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218 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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119 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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Exercise on partially ordered sets from Kunen's *Set Theory*

This problem is Exercise (F4) from Chapter VII of Kunen's text. Apparently, it is a result of Tarski. It goes as follows: Let $ \mathbb{P} $ be a partial order. Define $ \text{c.c.}(\mathbb{P}) $ ...
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283 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
367 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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3answers
357 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
130 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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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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375 views

List of explicit enumerations of rational numbers

A well-known mathematical fact is that the rational numbers are countable, i.e. there is a bijective function $$f:\mathbb{N}\rightarrow \mathbb{Q}$$ I am interesting in making a list of all explicit ...
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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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139 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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208 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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406 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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94 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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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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681 views

What infinity is greater than the continuum? Show with an example

The diagonal argument establishes that the continuum is greater than countable infinity. What is an example of the next infinity (or any greater infinity) and how can it be shown that there is no 1:1 ...
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280 views

How many cardinals are there?

I'm trying to do the following exercise: EXERCISE 9(X): Is there a natural end to this process of forming new infinite cardinals? We recommend this exercise instead of counting sheep when you have ...
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cardinality of all real sequences

I was wondering what the cardinality of the set of all real sequences is. A random search through this site says that it is equal to the cardinality of the real numbers. This is very surprising to me, ...
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The real line has cardinality at most $\aleph_2$, but transfinite ordinal space has arbitrarily high cardinality: what is wrong?

In the context of supertasks, people and mathematicians are comfortable with the idea of transfinite ordinal time, that is, that time can be divided into an arbitrarily high number of steps. In most ...
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Which set is unwell-orderable?

In textbook it says that every well-orderable set is equipotent to an initial ordinal number. However, of course unwell-orderable cannot equipotent to any ordinal number, but is there any set is ...
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159 views

An easy to understand definition of $\omega_1$?

I have two things I'm not sure in 100% about them. The first, is $\omega_1$. I have a little "feeling" of it, but if I'll be asked to define it - I don't know where to begin from. Perhaps it is ...
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How much is ${\aleph_0}^{\aleph _ 0}$? [duplicate]

How much is ${\aleph_0}^{\aleph _ 0}$? On the left I can find ${2}^{\aleph_0}\le {\aleph_0}^{\aleph _ 0}$ but on the right I can not found someone that is $\le$. In general, how do I use ...
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4answers
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Cardinality of the Irrationals [duplicate]

Possible Duplicate: Proof that the irrational numbers are uncountable We previously proved that $\mathbb{Q}$, the set of rational numbers, is countable and $\mathbb{R}$, the set of real ...
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282 views

Hilbert's Hotel and Infinities for Pre-university Students

Hilbert's paradox of the grand hotel is a fun and exciting ground to base a talk on the set theoretic concept of infinity for interested students - even in middle- and high school. However, it does ...
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Cardinality of Vitali sets: countably or uncountably infinite?

I am a bit confused about the cardinality of the Vitali sets. Just a quick background on what I gather about their construction so far: We divide the real interval $[0,1]$ into an uncountable number ...
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555 views

What is the cardinality of the group of bijections from $\Omega$ to $\Omega$ with finite support?

These questions cropped up in the discussion in this question, What is the cardinality of the group of bijections from $\Omega$ to $\Omega$ with finite support, where $\Omega=\mathbb{N}$? ...
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2answers
228 views

Ordinals that satisfy $\alpha = \aleph_\alpha$ with cofinality $\kappa$

I am trying to solve the following question: Prove that for every regular cardinal, $\kappa \gt \aleph_0$, there is a exists an $\alpha$ with cofinality $\kappa$ such that $\alpha = ...
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$\sum_{\alpha<\omega_2}|\alpha|^{\aleph_0}=\aleph_2\cdot\aleph_1^{\aleph_0}$

I've seen this statement in multiple posts (e.g. here and here), but I can't seem to understand it. I can see why $$\sum_{\alpha<\omega_2}|\alpha|^{\aleph_0}=\aleph_1^{\aleph_0},$$ by noting that ...
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203 views

Cardinal power $\kappa^\kappa$. When is it equal to $2^\kappa$?

Under what assumptions on an infinite cardinal $\kappa$ we have $$\kappa^\kappa= 2^\kappa?$$ Please delete this question. I know the answer.
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3answers
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Is there an infinite countable $\sigma$-algebra on an uncountable set

Let $\Omega$ be a set. If $\Omega$ is finite, then any $\sigma$-algebra on $\Omega$ is finite. If $\Omega$ is infinite and countable, a $\sigma$-algebra on $\Omega$ cannot be infinite and ...
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Conclusion about cardinalty.

Assume that: $$\left| T \right| > {\aleph _0}$$ Why can't one assume immediately that: $$\left| T \right| \cdot \left| T \right| > \left| T \right| \cdot {\aleph _0}$$
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The regularity of successor cardinal

I was looking at two different proofs of the fact that successor cardinals are regular. It struck me as odd that both proofs used AC. Looking at the concepts involved in defining cofinality I feel as ...
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162 views

Cardinal arithmetic questions

I have problem to solve: Let $a,b$ and $c$ be cardinal numbers. Prove that $a+b=b$, $b \le c$ implies $a+c=c$. And trying to prove this I got couple questions: For infinite cardinal $c$, is ...
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441 views

When an infinite union of countable sets is uncountable?

If I have $\kappa$ countable sets, when their union is not countable? only if $\kappa$ is uncountable? Using AC make differences?