This tag is for questions about cardinals and related topics such as cardinal arithmetics, clubs, stationary sets, cofinality, and principles such as $\lozenge$. Do not confuse with [large-cardinals] which is a technical concept about strong axioms of infinity.

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How many $p$-adic numbers are there?

Let $\mathbb Q_p$ be $p$-adic numbers field. I know that the cardinal of $\mathbb Z_p$ (interger $p$-adic numbers) is continuum, and every $p$-adic number $x$ can be in form $x=p^nx^\prime$, where ...
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1answer
135 views

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
498 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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1answer
407 views

Cofinality of cardinals

I have read some of the related questions (cofinality and its consequences, for example) about cofinality, but I still have no idea how to calculate the cofinality of an aleph. So, let's say I have ...
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2answers
101 views

$\sum_{\alpha<\omega_2}|\alpha|^{\aleph_0}=\aleph_2\cdot\aleph_1^{\aleph_0}$

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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3answers
407 views

Easiest way to prove that $2^{\aleph_0} = c$

$\aleph_0$ is the cardinality of the set of natural numbers, $\aleph_0 = |N|$. $c$ is the cardinality of the continuum, i.e. the set of real numbers $c = |R|$. I know that $|P(A)| = 2^{|A|}$. This ...
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4answers
426 views

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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1answer
191 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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4answers
1k views

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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4answers
217 views

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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2answers
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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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2answers
168 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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3answers
134 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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2answers
149 views

How badly can GCH fail?

I'm interested in "how many" cardinals we can cram between $\kappa$ and $2^\kappa$ before ZFC says "nope, too many" as measured by order type. So my question is (and i don't know if this is ...
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3answers
328 views

Proving that for infinite $\kappa$, $|[\kappa]^\lambda|=\kappa^\lambda$

Initially assume ZFC. Let $\binom{\kappa}{\lambda}$ denotes $\left|[\kappa]^{\lambda}\right|$ where $[\kappa]^{\lambda}$ is the collection of all subsets of $\kappa$ with cardinality $\lambda$. That ...
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3answers
263 views

The cardinality of a countable union of sets with less than continuum cardiality

Can continuum be a countable union of sets with cardinality, less than continuum? I can prove it for a finite union by mathematical induction from this: $\mathbb R = A_1 + A_2 => |\mathbb R| = ...
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1answer
152 views

the set of countable sets of Real numbers

I would like to ask some hints towards the proof that The set of countable sets in $\mathbb{R}$ is equinumerous to the set $\mathbb{R}$
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499 views

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
262 views

How to divide aleph numbers

Recently, I was wondering how division of aleph numbers would work. First, I thought about how finite cardinality division would work. What I came up with was that the result of $A/B$ where $A$ and ...
4
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1answer
147 views

Prove that: $\aleph_0 \cdot \frak{c} = \frak{c} \cdot \frak{c}$

I've been fiddling with this enough. Found an answer here but didn't quite understand it. How do I prove that: $$\aleph_0 \cdot \frak{c} \leq \frak{c} \cdot \frak{c}$$ $$\frak{c} \cdot \frak{c} \leq ...
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2answers
128 views

CH imples the existence of a function

I was studying an article and the author stated that CH implies that there exists a function from $\omega_1 \setminus \omega$ onto the set of all countable subsets of $\omega_1$ such that for each ...
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1answer
63 views

If $cf(\kappa)=\lambda$, then is every sequence of length $\lambda$ cofinal in $\kappa$?

Take $\omega_1$ for instance. Let's say I have a sequence of (distinct) ordinals of length $\omega_1$. Will this sequence be cofinal in $\omega_1$?
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3answers
147 views

$\aleph_1\leq A$ for an uncountable set $A$

So we have to prove (axiom of choice is given) that $\aleph_1$ which is the next aleph after $\aleph_0=\omega$ satisfies $$\aleph_1\leq A$$ given that $A$ is uncountable. Here was my reasoning. The ...
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1answer
111 views

Bijection between countable ordinals and reals

The set of all countable ordinals is $\omega_1$, which has a cardinality of $\aleph_1$. When accepting the continuum hypothesis, $2^{\aleph_0} = \aleph_1$, so a bijection between countable ordinals ...
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1answer
75 views

If $X$ is Hausdorff and $|X|> \mathfrak{c}$, does $X$ always have a uncountable discrete subspace?

Let $X$ be a Hausdorff topological space with $|X|> \mathfrak c$. Does $X$ always have a uncountable discrete subspace? Thanks for your help.
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1answer
72 views

A cardinal $\kappa$ such that $2^{\lambda}<\kappa$ for all $\lambda<\kappa$ is regular?

A cardinal $\kappa$ such that $2^{\lambda}<\kappa$ for all $\lambda<\kappa$ is regular? I would appreciate very much an answer
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2answers
1k views

Cartesian Product of Two Countable Sets is Countable

How can I prove that the Cartesian product of two countable sets is also countable?
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1answer
284 views

Cardinal Arithmetic versus Ordinal Arithmetic

I am reading Philosophy, not Set Theory, so please excuse the naivety of my question. My question concerns the wildly different character of ordinal arithmetic versus cardinal arithmetic. The ...
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2answers
93 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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2answers
79 views

About a proof of “$\bigcup A$ is a limit cardinal”

Assume that if $A$ is a set of cardinals such that $A$ contains no largest element and assume that we have shown that $\bigcup A$ is a cardinal. Now we want to show that $\bigcup A$ is a limit ...
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2answers
193 views

how to prove the addition of transfinite cardinal numbers?

How do you prove the following transfinite cardinal addition?: $ \alpha + \beta = \max(\alpha,\beta)$? And as the consequence, $\alpha + \alpha = \alpha$ where $\alpha$ and $\beta$ are transfinite ...
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2answers
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Proving the Cantor Pairing Function Bijective

How would you prove the Cantor Pairing Function bijective? I only know how to prove a bijection by showing (1) If $f(x) = f(y)$, then $x=y$ and (2) There exists an $x$ such that $f(x) = y$ How would ...
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2answers
259 views

Fodor's lemma on singular cardinals

Fodor's lemma asserts that if $\kappa$ is a regular and uncountable cardinal, then if $f(\alpha)<\alpha$ for a stationary subset of $\kappa$, then it is constant on stationary subset. Suppose ...
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2answers
222 views

If $|X|=|Y|=|X-Y|=\kappa$, can we find a bijection on $X$ that fixes $Y$ only?

in a previous question, I mistakenly attempted to subtract one cardinal number from another. Anyway, this got me to thinking, suppose I have two sets $X$ and $Y$, with $Y\subseteq X$. Suppose also ...
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1answer
77 views

How to prove that $\omega_1 \leq \mathfrak p$

I am trying to understand the stated above Theorem. $\mathfrak p$ is the smallest cardinality of any family $\mathcal F \subseteq [\omega]^\omega$, which has the strong finite intersection property, ...
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2answers
230 views

Every infinite subset of a countable set is countable.

Here is the proof I tried to weave while trying to prove this theorem: Theorem. Every infinite subset of a countable set is countable. Proof. Let $A$ be a countable set and $E\subset A$ be ...
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3answers
350 views

Applications of cardinal numbers

I know basic things about cardinality (I'm only in High School) like that since $\mathbb{Q}$ is countable, its cardinality is $\aleph_0$. Also that the cardinality of $\mathbb{R}$ is $2^{\aleph_0}$. ...
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2answers
102 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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3answers
395 views

The cardinality of the set of countably infinite subsets of an infinite set

Let $A$ be a set with card($A$)=$a$. What is the cardinal number of the set of countably infinite subsets of $A$? I see that this problem is equivalent to finding the cardinal number of the set of ...
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2answers
150 views

singular cardinals of every regular cofinality

First of all a couple of definitions which might be different to the standard ones: A function is called cofinal if $$f:\alpha\mapsto \beta$$is such that $$\sup\{ f(\gamma):\gamma<\alpha\}=\beta$$ ...
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2answers
344 views

Let $X$ and $Y$ be countable sets. Then $X\cup Y$ is countable

Since $X$ and $Y$ are countable, we have two bijections: (1) $f: \mathbb{N} \rightarrow X$ ; (2) $g: \mathbb{N} \rightarrow Y$. So to prove that $X\cup Y$ is countable, I figure I need to define ...
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1answer
73 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
129 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
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Proving that $\sf Add$$(\aleph_\omega , 1)$ collapses cardinals $\leq \aleph_\omega$

First, let me fix some notation. $\sf Fn$$(I, J, \kappa) = $ the poset of all partial functions $p$ such that $|p| < \kappa$, dom$(p) \subseteq I$ and rng$(p) \subseteq J$. $\sf Add$$(\kappa, ...
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2answers
207 views

Factorial of Infinite Cardinal

I have been thinking about the following problem: Let $A$ be a set of cardinality $k$ and denote $\sum_A$ the set of all bijection from $A$ to $A$. Also denote $k! = \mathrm{card}\left(\sum_A ...
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1answer
130 views

About cardinalities of almost disjoint systems of functions

Let us call a system $\{f_i; i\in I\}$ of functions $f_i\colon\omega\to\omega$ almost disjoint if the set $\{n\in\omega; f_i(n)=f_j(n)\}$ is finite whenever $i\ne j$. (I am not entirely sure whether ...
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1answer
68 views

GCH implies that $2^{<\kappa}=\kappa$

If GCH holds, then $ 2^{<\kappa}=\kappa$ for all $\kappa$ It is true that $2^{<\kappa}=\sup_{\delta <\kappa}(2^\delta)$ some explain this for me. Thanks in advance
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2answers
203 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
117 views

Is this theory $\kappa$-categorical when $\kappa$ is infinite and not $\le \aleph_0$?

Let $\mathcal L$ be a language with only a binary predicate $E$, and let $T$ be a theory of structures in which $E$ is an equivalence relation which partitions the structure into two infinite ...