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.

learn more… | top users | synonyms (1)

4
votes
2answers
286 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 ...
0
votes
1answer
735 views

Is the set of integer coefficient polynomials countable? [duplicate]

Possible Duplicate: Is the set of polynomial with coefficients on $\mathbb{Q}$ enumerable? The set of integer coefficient polynomials are countable, when the cardinality of each set of ...
5
votes
1answer
148 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
votes
1answer
800 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 ...
5
votes
3answers
229 views

Question on a proof from Jech's Set Theory of: If $X$ is a set of cardinals, then $\sup X$ is a cardinal.

The proof: Let $\alpha= \sup X$. If $f$ is a bijective mapping of $\alpha$ onto some $\beta < \alpha$, let $\kappa \in X$ be such that $\beta < \kappa \le \alpha$. Then $|\kappa|=|\{f(\xi): \xi ...
2
votes
2answers
134 views

A proof for the equality $\aleph_0 card(X) = card(X)$ with $X$ an infinite set?

Good evening, I want to show that all bases of a vector space have the same cardinality, and it needs the following equality : Let $\aleph_0$ be the cardinality of $\mathbb{N}$ and $X$ an infinite ...
6
votes
3answers
1k views

Number of countable subsets of $\mathbb{R}$

More generally, if a set $S$ has cardinality $\mathfrak{m}$, how many of its subsets have cardinality $\mathfrak{n}$? Clearly there are at least $2^\mathfrak{n}$ such subsets. I don't see how many ...
1
vote
3answers
603 views

How to prove that the set of all countable ordinals, $\omega_1$, is uncountable / has the cardinality $\aleph_1$?

The only reasoning I've seen given for this is that it's uncountable because it can't include itself an element. I'm a little unconvinced and was looking for a more proper formal proof demonstrating ...
2
votes
2answers
357 views

What is the cardinality of the set of functions having finite image?

If we are given two infinite sets $X$ and $Y$ we can consider the set $S$ of all functions from $X$ to $Y$, which has cardinality $|Y|^{|X|}$. Also, we can consider the set $F$ of all functions from ...
6
votes
1answer
485 views

On cardinality of equivalence classes

Say $R$ is the set of all Riemann integrable functions $f:[a,b]\rightarrow \mathbb{R}$. We define an equivalence relation in $R$ as follows: $f$ and $g$ are said to be integrally equivalent iff they ...
5
votes
1answer
244 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 ...
36
votes
7answers
3k views

Why is the Continuum Hypothesis (not) true?

I'm making my way through Thomas W Hungerfords's seminal text "Abstract Algebra 2nd Edition w/ Sets, Logics and Categories" where he makes the statement that the Continuum Hypothesis (There does not ...
3
votes
1answer
232 views

How many weak/strong limit cardinals exist under different assumptions?

I am trying to hastily answer the question in the title because I need it for another problem. I have been consulting multiple sources and am confused about the standard meanings of the following ...
5
votes
3answers
397 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 ...
0
votes
1answer
227 views

If $\kappa$ is a cardinal, $\aleph$ is any $\aleph$-number, and if $\kappa\leq\aleph$ then $\kappa$ can be well ordered as well.

I'm having trouble understanding the statement: If $\kappa$ is a cardinal, $\aleph$ is any $\aleph$-number, and if $\kappa\leq\aleph$ then $\kappa$ can be well ordered as well. I understand the ...
2
votes
1answer
109 views

Closing a subcategory under finite colimits by transfinite induction

Let $\mathcal{C}$ be a locally small category with all finite colimits, and let $\mathcal{A}$ be a small full subcategory. I wish to prove the following: Proposition. There exists a full subcategory ...
5
votes
2answers
348 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: ...
4
votes
3answers
661 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 ...
0
votes
1answer
94 views

If $a, p$ are cardinals satisfying $2 p = p$ and $a+p=2^p,$ then $a \ge 2^p.$

Theorem: If $\mathfrak a$ and $\mathfrak p$ are cardinals satisfying $2\mathfrak p = \mathfrak p$ and $\mathfrak a + \mathfrak p=2^\mathfrak p$, then $\mathfrak a \ge 2^\mathfrak p$. Here's a ...
7
votes
1answer
438 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 ...
4
votes
1answer
118 views

Proving $|A^A|=|2^A|$ for infinite $A$. [duplicate]

Possible Duplicate: Cardinal power $\kappa^\kappa$. When is it equal to $2^\kappa$? How can one prove that $|A^A|=|2^A|$ for infinite $A$? (summary of proof or providing link with proof ...
6
votes
2answers
2k views

How do you prove the trichotomy law for cardinal numbers?

Law of trichotomy is that for any two cardinals $a$ and $b$, exactly one of the conditions $a<b$, $a=b$, or $a>b$ holds.
8
votes
2answers
3k views

Why do the rationals, integers and naturals all have the same cardinality?

So I answered this question: Are all infinities equal? I believe my answer is correct, however one thing I couldn't explain fully, and which is bugging me, is why the rationals $\mathbb Q$, integers ...
3
votes
1answer
188 views

When collapsing a cardinal, what ordinal does it become?

Work in $V$. Let $P = \text{Col}(\omega, \omega_1)$ and suppose that $G$ is generic for $P$ over $V$. Then $V[G]\models |\omega_1^V|=\aleph_0$ and $\omega_2^V=\aleph_1$. In particular, ...
2
votes
1answer
220 views

Cardinal exponentiation problem from Halmos' Naive Set Theory

In chapter 24 of Halmos' Naive Set Theory the following problem is posed as an exercise (page 96): Prove that if $a, b$ and $c$ are cardinal numbers such that ${a}\le{b}$, then $a^c\le{b^c}$. ...
-3
votes
1answer
96 views

Rational numbers and cardinality of some subset set of them.

Let $G$ be the set of rational numbers of the form $m/n$ , where $m,n$ are positive integers and $n \leq g $ for some possitive integer $g$. Suppose it is bounded by $1/k$ , k is a positive integer ...
6
votes
2answers
163 views

Sum of cardinals without AC

Let $A$ and $B$ be infinite sets. To show $|A\cup B|=\max\{|A|,|B|\}$ we need AC. Now let us assume $|A|<|B|$. Can we show $|A\cup B|=|B|$ without AC?
0
votes
1answer
320 views

Is cardinality a total order? Is AC necessary? [duplicate]

Possible Duplicate: Is the class of cardinals totally ordered? Intuitively, it seems like for any sets $A,B$ either $\lvert A\rvert\leq \lvert B\rvert$ or $\lvert B \rvert \leq \lvert ...
2
votes
1answer
950 views

Cardinality of cartesian square

Given an infinite set $A$ - does the cardinality of $A$ equal to the cardinality of $A^2$?
0
votes
4answers
3k views

Why is the cardinality of irrational numbers greater than rational numbers?

This was asked by blogegog on a YouTube comment (gasp!): [Regarding Cantor's diagonal argument:] Couldn't I just make the same statement about rational numbers and say, 'take the largest ...
4
votes
1answer
118 views

Show that for $n, m \in \omega$, the ordinal and cardinal exponentiations $n^m$

This is an exercise from Kunen's book. Show that for $n, m \in \omega$, the ordinal and cardinal exponentiations $n^m$ are equal. What I've tried: I want to prove by using induction on $m$. ...
5
votes
2answers
910 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 $X$, ...
2
votes
2answers
62 views

Cocountable fibers

Let $C$ be an uncountable set. Can we construct a set $A \subseteq C^2$ such that it has a cocountable number of cocountable horizontal fibers, and a cocountable number of countable vertical fibers?
7
votes
1answer
422 views

What is the standard proof that $\dim(k^{\mathbb N})$ is uncountable?

This is my (silly) proof to a claim on top of p. 54 of Rotman's "Homological algebra". For $k$ an infinite field (the finite case is trivial) prove that $k^\mathbb{N}$, the $k$-space of functions ...
0
votes
2answers
2k views

Cardinality of a union of uncountable and countable set.

Suppose you have a set A which has the same cardinality with real numbers R, which means |A| = |R|. Also suppose that you have a finite set B , which of course has finite cardinality. Also suppose A ...
8
votes
1answer
482 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 ...
0
votes
1answer
121 views

How to prove that $2^\omega=\mathfrak{c}$?

Let $\mathfrak{c}$ denote the continuum. My textbook says that $2^\omega=\mathfrak{c}$. How can one prove this equality? Thanks ahead:)
1
vote
2answers
436 views

Non-aleph infinite cardinals

I'm now confused with a concept of $\aleph$. 1.$\aleph$ is a cardinal number that is well-ordered in ZF.(Defined as an initial ordinal that is equipotent with). Does that mean $\aleph_x$ in ZF may ...
3
votes
2answers
380 views

Cardinality of the complex numbers in ZF

As you all know, cardinality of $\mathbb{R} = 2^{\aleph_0}$ can be proved in ZF, since cardinality of $\mathbb{N} \times \mathbb{N} = \aleph_0$ can be proved in ZF. I know that the statement 'For any ...
2
votes
1answer
141 views

$\kappa$-complete, $\lambda$-saturated ideal properties

Kunen, II.56. Having trouble proving the properties of the following: The definition: $S(\kappa,\lambda,\mathbb{I})$ is the statement that $\kappa > \omega$ and $\mathbb{I}$ is a $\kappa$-complete ...
0
votes
2answers
89 views

About $|\operatorname{Sym}(\Omega)|$ when $\Omega$ is an infinite set.

Here is a problem: Show that if $\Omega$ is an infinite set, then $|\operatorname{Sym}(\Omega)|=2^{|\Omega|}$. I have worked on a problem related to a group that is $S=\bigcup_{n=1}^{\infty } ...
6
votes
1answer
413 views

Is there a way to define the “size” of an infinite set that takes into account “intuitive” differences between sets?

The usual way to define the "size" of an infinite set is through cardinality, so that e.g. the sets $\{1, 2, 3, 4, \ldots\}$ and $\{0, 1, 2, 3, 4, \ldots\}$ have the same cardinality. However, is this ...
5
votes
2answers
549 views

In set theory, what does the symbol $\mathfrak d$ mean?

What's meaning of this symbol in set theory as following, which seems like $b$? I know the symbol such as $\omega$, $\omega_1$, and so on, however, what does it denote in the lemma? Thanks ...
1
vote
1answer
151 views

Number of natural and real numbers [duplicate]

Possible Duplicate: The simplest way of proving that $|\mathcal{P}(\mathbb{N})| = |\mathbb{R}| = c$ I was reading Rubin and came across the fact that $2^{\aleph_0}$ is the cardinality of ...
3
votes
2answers
93 views

Question about cardinals without GCH

Without assuming the Generalized Continuum Hypothesis, how to show that there exists a uncountable cardinal $\kappa$ such that, for every $\lambda < \kappa$, one have $2^\lambda < \kappa$. With ...
7
votes
1answer
335 views

Number of well-ordering relations on a well-orderable infinite set $A$?

Given a well-orderable infinite set $A$, can we always say that the set $$\left\{R\in A\times A:\langle A,R\rangle\, \text{is a well-ordering}\right\}$$ has cardinality $2^{|A|}$? How much Choice is ...
2
votes
2answers
447 views

cardinality: The cardinality of the set of all relations over the natural numbers.

I have to find the cardinality of the set of all relations over the natural numbers, without any limitations. It seems to be א, but I can't find a function/other way to prove it. help anyone? ...
3
votes
4answers
406 views

Proving that there are infinite cardinal numbers >$\mathfrak{c}$

I was reading Simmons' book and he states that there are infinite cardinal numbers > $\mathfrak{c}$ where $\mathfrak{c}$ denotes the number of Real Numbers. For this, he states that we can construct ...
-1
votes
1answer
148 views

cardinality of possible prime decompositions, countably infinite bijection

Let $A = \{ p_{i},p_{i+1},\ldots,p_{n}\}$ be any finite set of prime numbers, where $i,n \in \mathbb{N}$. And $p_{i}\in A$, is the $i$th prime number i.e. $p_{2} = 3$. Let all possible finite sets ...
2
votes
3answers
598 views

cardinality of the set of countable partitions of $\mathbb{R}$

What is the cardinality of the set $$A=\{ P| P\ \text{is a countable partition of the reals} \}$$ ? I am searching on this for a while. I think the cardinality is $2^w$ where $w$ is the cardinality ...