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)

6
votes
3answers
634 views

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

Can we even find examples of infinity in nature?
6
votes
3answers
869 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
469 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
637 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
3answers
1k 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?
6
votes
2answers
767 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
259 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
3answers
454 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 \implies |\mathbb R| = ...
6
votes
2answers
56 views

Let $X \subseteq \mathbb R$ and $X$ has same cardinality as $\mathbb R$ , does there always exist a continuous surjection from $\mathbb R$ onto $X$ ?

Let $X \subseteq \mathbb R$ and $X$ has same cardinality as $\mathbb R$ , does there always exist a continuous surjection from $\mathbb R$ onto $X$ ? ( I know that there need not always be a ...
6
votes
1answer
129 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
3answers
712 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 ...
6
votes
2answers
156 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
4answers
884 views

Problems about Countability related to Function Spaces

Suppose we have the following sets, and determine whether they are countably infinite or uncountable . The set of all functions from $\mathbb{N}$ to $\mathbb{N}$. The set of all non-increasing ...
6
votes
3answers
748 views

Cardinality != Density?

I was in a discussion where I argued that the density of two sets of the same cardinality could be different in respect to the infinite range of non-negative integers. Does cardinality imply that any ...
6
votes
3answers
197 views

Prove: $B=\{x\in \Bbb R\mid x^2\in\Bbb Q\}$ is countable

Prove: The set $B=\{x\in \Bbb R\mid x^2\in\Bbb Q\}$ is countable I have this idea but I'm not sure if it's correct: We know $\Bbb Q$ is countable so we can list $\Bbb Q$ as $\Bbb Q= \{ ...
6
votes
3answers
139 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 ...
6
votes
2answers
287 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
216 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
410 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 ...
6
votes
1answer
1k views

Cardinality of a set that consists of all existing cardinalities

What is the easiest way to prove (if possible, without using ordinals etc. as my current math understanding of set theory counts only cardinals, and countable & uncountable sets) that the number ...
6
votes
2answers
100 views

Why was $\aleph$ (aleph) chosen for infinities?

Why did Cantor choose a letter from the Hebrew alphabet to represent infinities, rather than using some Greek letter?
6
votes
2answers
168 views

What is the value of $\beth_{\omega_1}^{\aleph_0}$?

It is well known that $\beth_\omega^{\aleph_0} = \beth_{\omega+1}$. This follows since for strong limit $\kappa$, we have $\kappa^\kappa = \kappa^{\mathrm{cf}(\kappa)}.$ Question. To the extent that ...
6
votes
2answers
148 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
3answers
2k 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 ...
6
votes
2answers
165 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
189 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
160 views

Prob 8, Sec 7 in Munkres' TOPOLOGY 2nd ed: How do we show these sets have the same cardinality?

Here's Prob. 8. Sec. 7 in Topology by James R. Munkres, 2nd edition: Let $X$ denote the two element set $\{0,1\}$; let $\mathscr{B}$ be the set of countable subsets of $X^{\omega}$. Show that ...
6
votes
2answers
193 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
2answers
216 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
1answer
230 views

Why continuum function isn't strictly increasing?

Is there any example that for cardinal numbers $\kappa < \lambda$, we have $2^\kappa = 2^\lambda$? My guess is that it only depends on whether GCH holds. Is it true?
6
votes
1answer
420 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
2answers
204 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
1answer
238 views

Well-foundedness of cardinals and the axiom of choice

Without axiom of choice it is not generally true that the class of all cardinal (in this question we consider Scott cardinal rather than cardinals as ordinals) is not well-founded under the ordinary ...
6
votes
2answers
132 views

Does $\sf GCH$ imply that every uncountable cardinal is of the form $2^\kappa$?

I think that this is a popular fallacy that GCH implies that every uncountable cardinal is of the form $2^\kappa$ for some $\kappa$. I think it does imply that for successor cardinals only. It cannot ...
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 ...
6
votes
1answer
123 views

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, ...
6
votes
1answer
105 views

$ZFC^- + AFA$ and infinite cardinals

$ZFC^-+AFA$ is a non-well-founded set theory, where $ZFC^-=ZFC-FA$ is $ZFC$ without the axiom of foundation, and $AFA$ is an anti-foundational axiom With the axiom of foundation we have that every ...
6
votes
1answer
88 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
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.
6
votes
1answer
128 views

Why do $2^{\lt\kappa}=\kappa$ and $\kappa^{\lt\kappa}=\kappa$ when the Generalized Continuum Hypothesis holds?

I'm going to assume GCH here. If that holds, then why do we have the equalities $2^{\lt\kappa}=\kappa$ for every $\kappa$, and $\kappa^{\lt\kappa}=\kappa$ for all regular $\kappa$?
6
votes
1answer
211 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
2answers
2k views

What is known about the power set of the real number line?

Cantor's theorem states that the cardinality of a set's powerset is strictly greater than that of the set itself. This clearly applies to the reals also; if I'm not mistaken, the cardinality of the ...
6
votes
2answers
732 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 ...
6
votes
1answer
458 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
507 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
155 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
68 views

What is the cardinality of an element of an free ultrafilter?

Let $U$ be a free ultrafilter on a set $X$. I want to prove that the cardinality of every element $u\in U$ is equipotent to $X$. Is that true? Or does it lack some hypothesis?
6
votes
1answer
786 views

List of explicit enumerations of rational numbers [closed]

A well-known mathematical fact is that the rational numbers are countable, i.e. there is a bijective function $$f:\mathbb{N}\rightarrow \mathbb{Q}$$ I am interesting in making a list of all explicit ...
6
votes
1answer
146 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
216 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 ...