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.

learn more… | top users | synonyms

6
votes
3answers
2k views

cardinality of set of all real continuous functions

Could somebody explain to me how to prove that the cardinality of all real continuous functions is $c$ ? The first problem is that I don't know how to show that each real continuous function $f: X ...
6
votes
2answers
287 views

prove cardinality rule $|A-B|=|B-A|\rightarrow|A|=|B|$

I need to prove this $|A-B|=|B-A|\rightarrow|A|=|B|$ I managed to come up with this: let $f:A-B\to B-A$ while $f$ is bijective. then define $g\colon A\to B$ as follows: $$g(x)=\begin{cases} ...
6
votes
2answers
443 views

What is the Cardinality of the Nameable Numbers?

Having just finished "Meta Math!" (Chaitin), I came across an interesting observation on infinite sets that I hadn't seen before. If I'm correct (and please let me know if I'm not): 1] There are ...
6
votes
2answers
834 views

How to understand the regular cardinal?

How to understand the regular cardinal? Could someone give me some examples?
6
votes
2answers
718 views

Cardinality of sets of subsets of $\mathbb{N}$

If we dont assume CH, is there a procedure to construct or define a set of subsets of $\mathbb{N}$ such that we cannot prove it to be of cardinality $\aleph_0$ or $\aleph_1$? Or if we assume not CH, ...
6
votes
1answer
240 views

is there a cardinality between the rational and the irrationals?

Was asked this question, and I have no idea about how to start proving it. Could someone give me some good reference material to start with.
6
votes
5answers
312 views

Are there more real numbers than we can actually imagine?

I mean, if we could imagine all the real numbers then we could assign each number a finite sentence (or a finite book). Since the set of the finite books is countable then the set of real numbers ...
6
votes
1answer
238 views

What is the cardinality of the set of all sequences in $\mathbb{R}$ converging to a real number?

Let $a$ be an real number and let $S$ be the set of all sequences in $\mathbb{R}$ converging to $a$. What is the Cardinality of $S$? Thanks
6
votes
1answer
107 views

What is the cardinality of $\Bbb{R}^L$?

By $\Bbb{R}^L$, I mean the set that is interpreted as $\Bbb{R}$ in $L$, Godel's constructible universe. For concreteness, and to avoid definitional questions about $\Bbb{R}$, I'm looking at the set ...
6
votes
2answers
187 views

Confusion about cofinality

I'm confused about the notion of the cofinality of a cardinal. Since I think the source of the confusion is the Von Neumann cardinal assignment, my first question is: Question 0. Is there an article ...
6
votes
2answers
193 views

Is it viable to ask in an infinite set about the Cardinality?

Can you ask given an infinite set about its cardinality? Does an infinite set have a cardinality? So, for example, what would be the cardinality of $+\infty$?
6
votes
2answers
2k 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 ...
6
votes
2answers
141 views

What is the product of finitely indexed alephs?

I'm simply curious about why the following equality holds: $ \displaystyle\prod_{n\lt\omega}\aleph_n=\aleph_\omega^{\aleph_0}. $ Much thanks!
6
votes
2answers
130 views

Skolem's paradox and models in set theory

Say you have a model $M$ (of $\mathsf{ZF}$) containing a set $S$ that you know is countable outside the model but the enumeration is missing from the model so that $S$ appears to be uncountable in ...
6
votes
2answers
142 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?
6
votes
3answers
135 views

Cardinality of a some linear ordering is at most that of a given cardinal?

This is an intuitive idea that I've used for a while, but don't know how to explain formally. Suppose $(A,\prec)$ is some linear ordering, and each initial segment of $A$ has cardinality strictly ...
6
votes
2answers
144 views

In ZF, how would the structure of the cardinal numbers change by adopting this definition of cardinality?

In ZFC, a good way of ordering sets by cardinality is by leveraging the notion of an injection. We define: $$X \lesssim Y \leftrightarrow \mbox{ there exists an injection } X \rightarrow Y.$$ ...
6
votes
1answer
318 views

Can an infinite cardinal number be a sum of two smaller cardinal number?

Let $\kappa$ be an infinite cardinal number. My question is whether there are $\lambda$ and $\mu$ such that both $<\kappa$ but $\lambda+\mu=\kappa$? If AC holds, then the answer is definitely ...
6
votes
2answers
176 views

What is the product of all nonzero, finite cardinals?

To be specific, why does the following equality hold? $$ \prod_{0\lt n\lt\omega}n=2^{\aleph_0} $$
6
votes
2answers
309 views

mahlo and hyper-inaccessible cardinals

Wikipedia states that a Mahlo cardinal is hyper-inaccessible, hyper-hyper-inaccessible, etc. Is this a characterisation of Mahlo? If not what about "alpha = hyper^alpha-inaccessible" with the obvious ...
6
votes
2answers
179 views

What happens after the cardinality $\mathfrak{c}$?

While having measure theory this year the following came in my mind: When we go from finite objects to infinite we "lose" a lot of properties. For example the summation isn't well defined ...
6
votes
2answers
188 views

Can proper classes also have cardinality?

In some set theories such as ZF+GAC, in which GAC is global axiom of choice, the Von Neumann universe $V$ bijects to $Ord$, the class of ordinals. It suggests us that proper classes may also have ...
6
votes
1answer
297 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 ...
6
votes
1answer
66 views

How to show $\kappa^{cf(\kappa)}>\kappa$?

For $\kappa \geq \omega$, which is a cardinal, then how to show $\kappa^{cf(\kappa)}>\kappa$? My idea: When $\kappa=\aleph_\omega$, then $cf(\kappa)=\omega$, is $\kappa^\omega>\kappa$? It seems ...
6
votes
1answer
221 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 ...
6
votes
1answer
158 views

Sequence of surjections imply choice

I am reading a paper where a side remark said that if a sequence $\langle g_\beta\colon\omega\to\beta\mid\beta<\omega_1\rangle$ is a sequence of surjections then $\omega_1$ is regular. I have ...
6
votes
1answer
135 views

How to prove that a regular cardinal cannot be expressed as a union of sets with less cardinality?

My question is about the following: Using the Axiom of Choice show that: If $\kappa\ge\omega$ is a regular cardinal, $\gamma\le\kappa$, and $\langle A_\alpha\mid\alpha\lt\gamma\rangle$ is a ...
6
votes
1answer
199 views

Question about the order of a Stationary subset of $ \kappa$

Greets I'm trying to prove one part of exercise 8.14 of Jech's "Set Theory", namely that if $o(k)\geq k$, then $k$ is weakly inaccessible, where $\kappa$ is regular; $o(\kappa)$ is defined as ...
6
votes
1answer
316 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 ...
6
votes
2answers
136 views

Effective cardinality

Consider $X,Y \subseteq \mathbb{N}$. We say that $X \equiv Y$ iff there exists a bijection between $X$ and $Y$. We say that $X \equiv_c Y$ iff there exist a bijective computable function between ...
6
votes
1answer
94 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 ...
6
votes
1answer
131 views

$\kappa <\operatorname{cf}(2^\kappa)$ without König's inequality

How can I prove $\kappa<\operatorname{cf}(2^\kappa)$ inequality without using König's inequality? We got this as a practice exercise, but I don't know how to approach this without König. Any hint ...
6
votes
1answer
197 views

How to explain that $\Bbb{R}$ is not countable to a non-mathematician

What is the best way to explain that $\Bbb{R}$ is not countable assuming that the audience is formed of people who are not mathematicians? I ask this because these days I'm in a debate with someone ...
6
votes
1answer
82 views

$V=L[A]$ implies GCH for $A\subset \aleph_1$

On page 14 of the introduction to Vol. II of Gödel's collected works: By a slightly more difficult argument one can show that GCH continues to hold if $V=L[a]$ and $a\subseteq\aleph_1$. Does ...
6
votes
1answer
354 views

Finding the cardinality of a set

I have to construct a few things before I get to my question: it's possible we don't need all of it, but I am stuck on the very last step so I should recap everything so far. Let $\kappa$ be a ...
6
votes
1answer
343 views

cardinality of infinite sets

prove or disprove: If two infinite sets $A$,$B$ have the same cardinality, then $A\cup B$ and $A$ have the same cardinality. I even cannot make a judgement. P.S: Can this be done without using ...
6
votes
1answer
348 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 ...
6
votes
0answers
110 views

AC iff $P(\delta)$ can be well-ordered for all $\delta\in{\bf On}$ [duplicate]

I'm trying to do the following exercise: Exercise. Assume ZF. Then AC is equivalent to the statement that $P(\delta)$ can be well-ordered for all $\delta\in{\bf On}$. I'm struggling with the ...
6
votes
0answers
227 views

About singular $\beth_{\alpha}$ for limit ordinals $\alpha$

Why are there cofinitely many regular cardinals of type $\beth_{\beta}$ below any given singular $\beth_{\alpha}$, if $\alpha$ is a limit ordinal?
6
votes
0answers
129 views

Is there an upper bound to the number of rings that can be obtained from a semigroup with zero by defining an additive operation?

Let $\mathscr S$ be the class of all semigroups with zero. For $(S,\times,0)\in\mathscr S,$ I want to count additive operations $+$ on $S$ such that $(S,+,\times,0)$ is a ring (possibly without ...
5
votes
3answers
553 views

Prove that transcendental numbers exist: Are there less paniful ways of doing it?

I've found this exercise on Boolos' Logic and Computability: A real number $x$ is called algebraic if it is a solution to some equation of the form: $$c_{\small d}x^{\small ...
5
votes
2answers
396 views

Why are infinite cardinals limit ordinals?

My book states this as obvious, but it isn't so trivial to me. thanks
5
votes
3answers
127 views

$|\mathbb{R}^\mathbb{R}|$ vs $|P(\mathbb{R})|$ Which is bigger?

$|\mathbb{R}^\mathbb{R}|$ vs $|P(\mathbb{R})|$ where $\mathbb{R}^\mathbb{R} =\{f | f:\mathbb{R} \rightarrow \mathbb{R}\}$ Are they equal? Which is bigger? How can I prove it?
5
votes
3answers
243 views

Question about Generalized Continuum Hypothesis

I wonder how the Generalized Continuum Hypothesis reveal that $A\times A$ is equivalent to $A$? $A$ is any infinite set.
5
votes
2answers
457 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 ...
5
votes
2answers
1k views

Cantor-Bernstein-like theorem: If $f\colon A\to B$ is injection and $g\colon A\to B$ is surjective, can we prove there is a bijection as well?

I've been trying to find this proof: If there exists $f \colon A\to B$ injective and $g \colon A \to B$ surjective, prove there exists $h \colon A \to B$ bijective. I thought of using ...
5
votes
3answers
298 views

Infinite sets with cardinality less than the natural numbers

Are there any infinite sets that have a lower cardinality than the natural numbers? Is there a proof of this?
5
votes
3answers
211 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 ...
5
votes
1answer
113 views

Is $\alpha^{\beta}=2^{\beta}$ for all infinite cardinalities $\alpha, \beta$?

Is $\alpha^{\beta}=2^{\beta}$ for all infinite cardinalities $\alpha, \beta$? I was wondering about this since I've encountered examples of times where this holds, but I can't seem to prove it myself ...
5
votes
4answers
730 views

What are Aleph numbers intuitively?

I cannot get my head around the concept of the `types' of Aleph infinity. What is an easy intuitive way to see when you are given the integer numbers $\aleph_0$ the $\aleph_1$ will follow?