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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0answers
109 views

Sets of Cardinals without choice

I have a theory that the axiom of choice is equivalent to the statement that sets of distinct cardinals are well ordered by cardinality. I can prove that the axiom of choice implies this. However I am ...
8
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3answers
915 views

The largest number system

If my number set construction memory doesn't fail me (I'll edit if errors are pointed out), we start out with Peano's axioms to get to $\mathbb{N}$, and in the need of an additive inverse for its ...
8
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4answers
2k views

Cardinality of all cardinalities

Let $C = \{0, 1, 2, \ldots, \aleph_0, \aleph_1, \aleph_2, \ldots\}$. What is $\left|C\right|$? Or is it even well-defined?
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2answers
1k views

Cardinality of the infinite sets

Consider the following problem: Which of the following sets has the greatest cardinality? A. ${\mathbb R}$ B. The set of all functions from ${\mathbb Z}$ to ${\mathbb Z}$ C. The ...
8
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4answers
731 views

What's the cardinality of all sequences with coefficients in an infinite set?

My motivation for asking this question is that a classmate of mine asked me some kind of question that made me think of this one. I can't recall his exact question because he is kind of messy (both ...
8
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3answers
756 views

Is the class of cardinals totally ordered?

In a Wikipedia article http://en.wikipedia.org/wiki/Aleph_number#Aleph-one I encountered the following sentence: "If the axiom of choice (AC) is used, it can be proved that the class of cardinal ...
8
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4answers
2k views

Do the real numbers and the complex numbers have the same cardinality?

So it's easy to show that the rationals and the integers have the same size, using everyone's favorite spiral-around-the-grid. Can the approach be extended to say that the set of complex numbers has ...
8
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2answers
458 views

Does taking the power set give you the “next biggest cardinal”

I know that if you take the power set of a set, it has a higher cardinality. Therefore there are an infinite number of them as $P^{n}(\mathbb{N})$(the nth power set of the naturals) Let's say $$C_{n} ...
8
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3answers
269 views

Why infinite cardinalities are not “dense”?

What tells us that the structure of the cardinals is "discrete"? I'm not using the words "discrete" and "dense" with their formal meanings. Maybe I have this confusion because I'm using concepts ...
8
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1answer
343 views

Is the axiom of choice needed to show that $a^2=a$?

A comment on this answer states that choice is needed for the statement that $a^2=a$ for all infinite cardinals $a$. In Thomas Jech's Set Theory (3rd edition), his theorem 3.5 proves this statement ...
8
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2answers
439 views

axiom of choice: cardinality of general disjoint union

I have this exercise involving the axiom of choice, but I don't understand where it's needed: Let $(X_i)_{i \in I}$ and $(Y_i)_{i \in I}$ be pairwise disjoint sets with $|X_i| = |Y_i|$. Prove, using ...
8
votes
3answers
167 views

Cardinality of the Union is less than the cardinality of the Cartesian product

Suppose I have two collections of sets, $\{A_i\}_{i \in I}$ and $\{B_i\}_{i \in I}$. It is given in the problem that $$\mathrm{card}(A_i) < \mathrm{card}(B_i) \text{ for all }i \in I $$ I want to ...
8
votes
2answers
205 views

Does $\mathcal P ( \mathbb R ) \otimes \mathcal P ( \mathbb R ) = \mathcal P ( \mathbb R \times \mathbb R )$?

Obviously the answer's yes if I replace $\mathbb R$ by a countable set. If $\mathbb R$ is replaced by a set $X$ having a greater cardinality, then the diagonal $\{ (x,x) , \ x\in X\}$ is not in the ...
8
votes
2answers
113 views

Can we conclude $\prod_{\kappa \in Crd, \kappa=1}^{\kappa<\aleph_\alpha}\kappa=2^{\aleph_\alpha}$ in ZFC?

In complex analysis, there is a function called Euler's Gamma function. Whenever given a positive integer $n+1$, it will return $n!=\prod_{i=1}^{i < n+1}i$. I'm not sure if there is similar ...
8
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2answers
221 views

Why does $\omega$ have the same cardinality in every (transitive) model of ZF?

As in the title: Why does $\omega$ have the same cardinality in every (transitive) model of ZF? I've been thinking about this for some time now. Can someone show me how to show this by showing me a ...
8
votes
2answers
89 views

Proof of non-existence of non-principal $\kappa$-complete ultrafilter

Let $\lambda$ be a cardinal. I would like to prove that for all cardinals $\lambda < \kappa \leq 2^\lambda$, there can't be a $\kappa$-complete non-principal ultrafilter on $\kappa$. Here is my ...
8
votes
1answer
445 views

Tightness and countable intersection of neighborhoods

The following is a problem a colleague has encountered. He would like to know whether the following conjecture is right, wrong, or neither: Let $X$ be a topological space of countable tightness ...
8
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1answer
159 views

Is the cardinality of uncountable $G_{\delta}$ set of $\mathbb{R}$ equals the cardinality of the continuum?

It is known that closed sets of $\mathbb{R}$ satisfies continuum hypothesis, that is, every closed subset of $\mathbb{R}$ is either countable or of the cardinality of the continuum. Is the ...
8
votes
1answer
418 views

For every $n < \omega$, $\aleph_n^{\aleph_0} = \max(\aleph_n,\aleph_0^{\aleph_0})$

I have proved that if $\aleph_n \leq \aleph_0^{\aleph_0}$, then $\aleph_n^{\aleph_0} \leq \max(\aleph_n,\aleph_0^{\aleph_0})$. Clearly $\aleph_n \leq \aleph_n^{\aleph_0}$ and $\aleph_0^{\aleph_0} \leq ...
8
votes
1answer
2k views

A simple bijection between $\mathbb{R}$ and $\mathbb{R}^4$ or $\mathbb{R}^n$?

How to form a bijection from $(0,1]$ to $\mathbb{R}$: $$f(x) = \left\{\begin{array}{ll} 2-\frac{1}{x}&\text{if }x\in(0, .5]\\ \frac{2x-1}{1-x}&\text{if }x\in(.5, 1]. \end{array}\right.$$ ...
8
votes
1answer
505 views

cardinal exponentiation, $k^{<\lambda}$

I have the following well-known exercise in cardinal arithmetics: If $\kappa, \lambda$ are cardinals such that $\lambda$ is infinite, then $\kappa^{<\lambda}$ equals the supremum of the ...
7
votes
4answers
1k views

Simple example of uncountable ordinal

Can you make a simple example of an uncountable ordinal? With simple I mean that it is easy to prove that the ordinal is uncountable. I know that the set of all the countable ordinals is an ...
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7answers
2k views

Partition of N into infinite number of infinite disjoint sets?

Is it possible to have a countable infinite number of countable infinite sets such that no two sets share an element and their union is the positive integers?
7
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2answers
252 views

A question about the cardinality of the set of all the bijections from $M$ to itself

$M$ is a set.We denote the set of all the bijections from $M$ to itself by $K$.Is the cardinality of $K$ larger than the cardinality of $M$?
7
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2answers
343 views

Implications of continuum hypothesis and consistency of ZFC

I've been a bit confused whilst doing some reading. I think the confusion arises because I am trying to read Wikipedia on topics without being able to work along. Anyway, I have read that, of course, ...
7
votes
2answers
344 views

What does Martin's Maximum imply for $P(\mathbb{R})$?

Prompted by this question: of course Godel's constructibility axiom implies that $P(S)$ is minimal for any set $S$ and so handily answers the question of the size of the power set of the continuum in ...
7
votes
1answer
274 views

What is $\aleph_0$ powered to $\aleph_0$?

By definition $\aleph_1 = 2 ^{\aleph_0}$. And since $2 < \aleph_0$, then $2^{\aleph_0} = {\aleph_1} \le \aleph_0 ^ {\aleph_0}$. However, I do not know what exactly $\aleph_0 ^ {\aleph_0}$ is or how ...
7
votes
3answers
110 views

What questions become answerable/computable given an uncountable character set?

Having reached the concluding portion of my first course in real analysis, one subject that I feel was not adequately addressed was the issue of cardinalities. This is a subject I was interested in ...
7
votes
1answer
142 views

Bijection between $\mathbb{Z}\times\mathbb{Z}\times\dots$ and $\mathbb{R}$

Is there a bijection between $\mathbb{Z}\times\mathbb{Z}\times\dots$ for countably infinitely many $\mathbb{Z}$'s and $\mathbb{R}$? That is, is $\mathbb{Z}\times\mathbb{Z}\times\dots$, repeated ...
7
votes
4answers
495 views

Fractional cardinalities of sets

Is there any extension of the usual notion of cardinalities of sets such that there is some sets with fractional cardinalities such as 5/2, ie a set with 2.5 elements, what would be an example of such ...
7
votes
1answer
424 views

What is the cardinality of a set of all monotonic functions on a segment $[0,1]$?

What is the cardinality of a set of all real monotonic functions on a segment $[0,1]$? Does it really matter that functions are monotonic?
7
votes
2answers
142 views

Is it possible that $2^n=3^n$ for some Dedekind-finite cardinal $n\gt0$?

Is it possible that $2^n=3^n$ for some Dedekind-finite cardinal $n\gt0$? I think the question speaks for itself, but let me try and satisfy the "quality standards" algorithm by padding it. Yes, I ...
7
votes
1answer
46 views

Does deleting a subset of an infinite set that has strictly smaller cardinality leave the cardinality of the infinite set unchanged?

Is it a theorem of ZFC that if $X \subseteq Y$, $|X| < |Y|$, $Y$ is infinite, then $|Y \setminus X|=|Y|$?
7
votes
1answer
273 views

Cardinal equality question

The question contains 2 stages: 1. Prove that ${\aleph_n} ^ {\aleph_1} = {2}^{\aleph_1}\cdot\aleph_n$ This one is pretty clear by induction and by applying Hausdorf's formula. 2. Prove ...
7
votes
1answer
136 views

What would a world where $\mathsf{CH}$ is false look like?

My question is a little more specific than the title may lead to believe. In the article The set-theoretic multiverse (J.D. Hamkins), the author writes the following: [...] the continuum is ...
7
votes
1answer
318 views

powers of singular cardinals

I am trying to solve the following two problems: 1) if $\beta <\omega_1$, $2^{\aleph_1}<\aleph_{\omega_1}$, and $\aleph_\alpha^{\aleph_0} \leq \aleph_{\alpha +\beta}$ for a stationary set of ...
7
votes
1answer
149 views

weak consequence of GCH

Can ZFC prove that there is a regular uncountable cardinal $\kappa$ such that $2^{<\kappa} < 2^\kappa$? Note, if the answer is no, it would require a strong global violation of SCH, so large ...
7
votes
1answer
89 views

Characterizing categories by size

Usually one distinguishes five classes of categories by size, and there are examples for all of them: finite categories locally finite categories small categories locally small categories large ...
7
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2answers
204 views

Cardinality of the set of open functions?

I was wondering what is the cardinality of the set $ \{ f : \mathbb{R} \to \mathbb{R} \mid f \text{ is open } \} $ (i.e., $f(U) \subseteq \mathbb{R}$ is open for all open $U$). There are at least $c = ...
7
votes
1answer
112 views

Why should the union of such a family of sets have cardinality $\aleph_1?$

Let $\mathcal A$ be a family of infinite countable sets which is linearly ordered by inclusion and such that $\bigcup\mathcal A$ is uncountable. I have to prove that $|\bigcup\mathcal A|=\aleph_1.$ I ...
7
votes
1answer
724 views

Examples of sets whose cardinalities are $\aleph_{n}$, or any large cardinal. (not assuming GCH)

One of the answers to this question indicates that large cardinals are useful for destructive testing of set theory. That aside, and not assuming GCH, are there any sets known that have a cardinality ...
7
votes
1answer
102 views

Constructing a Banach space of cardinality $\beth_{\omega+1}$

This is related to yesterday's question Constructing a vector space of dimension $\beth_\omega$; it's the next exercise (I.13.35 (a)) in Kunen's Set Theory. Let $B_0 = \ell^1$ and let $B_{n+1} = ...
7
votes
1answer
139 views

$(\beth_{\omega})^\omega=\beth_{\omega+1}$

I'm trying to show that $(\beth_{\omega})^\omega=2^{\beth_\omega}$. This is an exercise in Kunen where he suggests to encode subsets of $\beth_\omega$ with functions from ...
6
votes
3answers
676 views

How do we prove the existence of uncountably many transcendental numbers?

I know how to prove the countability of sets using equivalence relations to other sets, but I'm not sure how to go about proving the uncountability of the transcendental numbers (i.e., numbers that ...
6
votes
5answers
260 views

cardinality of the set of $ \varphi: \mathbb N \to \mathbb N$ such that $\varphi$ is an increasing sequence

I know that the set of functions $ f:\mathbb N \to \mathbb N$ is uncountable, but what if we consider only $f$ such that $f$ is increasing? I want to know if this set is countable D: and also the case ...
6
votes
4answers
1k views

What is the set of all functions from $\{0, 1\}$ to $\mathbb{N}$ equinumerous to?

What is the set of all functions from $\{0, 1\}$ to $\mathbb{N}$ equinumerous to? I have figured out the question when it's the other way around, but I am not making any progress here. The worst thing ...
6
votes
3answers
321 views

What is the first cardinal number which is grearter than continuum?

What is the first cardinal number which is grearter than continuum? We denote it by ? Thanks very much.
6
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3answers
537 views

Can we distinguish $\aleph_0$ from $\aleph_1$ in Nature?

Can we even find examples of infinity in nature?
6
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3answers
574 views

How to prove cardinality equality ($\mathfrak c^\mathfrak c=2^\mathfrak c$)

How do I prove this cardinality equality:$\mathfrak c^\mathfrak c=2^\mathfrak c$ I have failed to prove this after lots of trail - but I am certain it's true How can I prove this?
6
votes
2answers
273 views

Proving existence of a surjection $2^{\aleph_0} \to \aleph_1$ without AC

I'm quite sure I'm missing something obvious, but I can't seem to work out the following problem (web search indicates that it has a solution, but I didn't manage to locate one -- hence the ...