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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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 $f \colon X \to 2^X$ ...
5
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1answer
179 views

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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2answers
245 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: ...
5
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1answer
359 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
340 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 ...
5
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1answer
114 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 ...
5
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1answer
98 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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2answers
90 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
136 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., $Y$ ...
5
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1answer
193 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 ...
5
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1answer
365 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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0answers
64 views

Raising a partial function to the power of an ordinal

Consider a set $X$, and let $f : X \rightarrow X$ denote a partial function. Then for natural $n$, we can define $f^n$ as iterated composition, e.g. $f^2 = f \circ f$. Now suppose that $X$ is also ...
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0answers
175 views

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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5answers
269 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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3answers
258 views

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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4answers
389 views
4
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3answers
142 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 ...
4
votes
1answer
164 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
2k 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
251 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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4answers
510 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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3answers
294 views

What does $H(\kappa)$ mean?

As the typical references (Wikipedia, Mathworld, etc.) don't seem to address this satisfactorily, I figured this would be a good place to put a nice formal definition. Hence: I've heard that if ...
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4answers
533 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
194 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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2answers
108 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 ...
4
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1answer
198 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.
4
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4answers
219 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
129 views

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 ...
4
votes
3answers
152 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 ...
4
votes
2answers
153 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
364 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
92 views

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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1answer
182 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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2answers
618 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
297 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
151 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
133 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
66 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
151 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 ...
4
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1answer
121 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 ...
4
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1answer
76 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
77 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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3answers
694 views

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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2answers
95 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
81 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
223 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
2k views

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
227 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 ...
4
votes
1answer
39 views

Is there a cardinal $\kappa$ with uncountable cofinality such that $\kappa<\kappa^{\aleph_0}$?

Is there a cardinal $\kappa$ with uncountable cofinality such that $\kappa<\kappa^{\aleph_0}$? The question is motivated by the observation that $\kappa< \kappa^{{\rm cf}\kappa}$ for any ...
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1answer
95 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, ...