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
1answer
525 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
161 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 $X$...
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
150 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
218 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
227 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}) $ ...
6
votes
1answer
162 views

Dominating strategically $\omega_1$ reals

For a given $\kappa > \omega$, define the game $d(\kappa)$ that runs for $\omega$ stages as follows: At stage $n$, player I chooses a sequence of elements of $\omega$, $g_n$ of length $\kappa$, and ...
6
votes
1answer
421 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
92 views

Cardinality of order topology?

Just out of interest: The cardinality of the Euclidean topology on the real line is $c$. In general, if $X$ is totally ordered of cardinality $\alpha$, the order topology on $X$ must have cardinality $...
6
votes
1answer
437 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
0answers
53 views

How to solve probability when sample space is infinite?

I came up with a random problem yesterday: Suppose that in a random trial, each point $(x,y)$ where $x,y \in \mathbb{R}$ and $0 \leq x,y \leq 1$ is assigned a value of $0$ with 50% chance and a ...
6
votes
0answers
192 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 ...
6
votes
0answers
141 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 unity)....
5
votes
3answers
1k 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 d}+c_{\...
5
votes
3answers
511 views

Are there fields and ordered fields of every infinite cardinality?

On the top of my head, I cannot think of any fields of cardinality more than that of the reals. (It is known that the process of algebraic closure does not increase the cardinality of an infinite ...
5
votes
2answers
1k views

The Continuum Hypothesis & The Axiom of Choice

Does anyone here know of a reference to an analysis on a proposed relationship between The Continuum Hypothesis and The Axiom of Choice?
5
votes
2answers
849 views

Why are infinite cardinals limit ordinals?

My book states this as obvious, but it isn't so trivial to me. thanks
5
votes
5answers
337 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 ...
5
votes
3answers
385 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
3answers
528 views

Which set is unwell-orderable?

In textbook it says that every well-orderable set is equipotent to an initial ordinal number. However, of course unwell-orderable cannot equipotent to any ordinal number, but is there any set is ...
5
votes
3answers
129 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
548 views

How many $p$-adic numbers are there?

Let $\mathbb Q_p$ be $p$-adic numbers field. I know that the cardinal of $\mathbb Z_p$ (interger $p$-adic numbers) is continuum, and every $p$-adic number $x$ can be in form $x=p^nx^\prime$, where $x^\...
5
votes
2answers
556 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
2k 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
2answers
203 views

Is the cardinality of $\mathbb{Z^R}$=$\mathbb{R^Z}$?

Previously in this question, we have found that $\mathbb{R^Z}$ is uncountable and its multiset of components, denoted by $$K = \{ (..., 0, 0, w, 0, 0, ... ) : w \in \mathbb{R} \}$$ where for each ...
5
votes
2answers
532 views

How can we show there is a set whose cardinality is greater than $\cal P^n(\Bbb N)$ for every natural number $n$?

I haven't studied properly the theory of infinities yet. Let $A_0$ denote the set of natural numbers. Let $A_{i+1}$ denote the set whose elements are all the subsets of $A_i$ for $i=0,...,n,...$ I ...
5
votes
5answers
384 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 ...
5
votes
3answers
455 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 $\...
5
votes
2answers
164 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 ...
5
votes
3answers
237 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
135 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
1answer
461 views

Intuition about the size of $\aleph_k$ with $k>1$

Assuming CH for simplicity, I know of some more or less intuitive way to think about difference in sizes of $\aleph_0$ and $\aleph_1$. The most straightforward is the distinction of natural/rational ...
5
votes
4answers
1k views

Cardinality of Irrational Numbers

I know and I have proved more than once that the set of irrational numbers ($\mathbb{I}$) is uncountable, but now I'm given to solve this problem: Show that $|\mathbb{I}|=|\mathbb{R}|$, How can I ...
5
votes
1answer
189 views

What is the value of $\aleph_1^{\aleph_0}$?

Is there any neat way to calculate the value of $\aleph_1^{\aleph_0}$?
5
votes
2answers
1k 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 $X$, ...
5
votes
1answer
86 views

Which has bigger cardinality, $\mathbb Q^\mathbb R$ or $\mathbb R^\mathbb Q$?

Which has bigger cardinality, $\mathbb Q^\mathbb R$ or $\mathbb R^\mathbb Q$ ? I think I should use the Schroder-Bernstein theorem but I can't find the necessary injections/ prove that there aren't ...
5
votes
1answer
170 views

Measure of an elementary set in terms of cardinality

In Terry Tao's textbook on measure theory and integration, he notes that, given an elementary set $A$, the length of $A$, denoted $|A|$, may be written discretely as $$|A| = \lim_{n \to \infty}\frac{...
5
votes
3answers
287 views

Uncountable Cardinals without AC

I am doing an exercise, proving that without AC or Replacement that there are uncountable cardinals. As a point of reference I looked at the proof in Kunen's "The Foundations of Mathematics" that ...
5
votes
5answers
190 views

$\mathrm{card} ( \mathbb{Q})=\mathrm{card}( \mathbb{Q^c})$: Overcoming Wrong Intuition

This is a widespread intuitive argument, asserting that $\mathrm{card} ( \mathbb{Q})=\mathrm{card}( \mathbb{Q^c})$: Between any two rational numbers there's an irrational one and vice versa. So $\...
5
votes
2answers
119 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 ...
5
votes
1answer
329 views

“Real” cardinality, say, $\aleph_\pi$?

Is there any meaningful definition to afford for $\aleph_r$ (as in cardinality) where $r\in\mathbb{R}^+$? $r\in\mathbb{C}$? What about $\aleph_{\aleph_0}$? Can we iterate this? $\aleph_{\aleph_{\...
5
votes
1answer
103 views

Cardinal Arithmetic: $\beth_\omega$

Is the following equality independent of ZFC: $$\bigcup_{\aleph_{\beta}<\beth_{\omega}}2^{\aleph_{\beta}}=\beth_\omega $$ Consistent: Now that the equality is consistent with ZFC since it holds ...
5
votes
1answer
106 views

Landing between $\beth_\lambda$ and $\beth_{\lambda+1}.$

Main Question. Is it consistent with ZFC that there exists a limit ordinal $\lambda$ and a cardinal number $\nu$ satisfying $$\beth_\lambda < \beth_\lambda^\nu < \beth_{\lambda+1}?$$ I am also ...
5
votes
1answer
152 views

Existence of a regular uncountable $\aleph_{\alpha}$ without $\mathsf{AC}$

Set theory (Jech) $\text{p.}\;27:$ It is an open problem whether one can prove without the axiom of choice that there exists a regular uncountable $\aleph_{\alpha}\;($the informed guess is that ...
5
votes
1answer
184 views

Injective or constant function on measurable cardinal

I'm working on the following question: Let $\kappa$ regular cardinal, and $f:\kappa \rightarrow \kappa$. Prove that $\{\alpha < \kappa\mid f|_\alpha : \alpha \rightarrow \alpha\}$ is a ...
5
votes
1answer
187 views

Aleph arithmetic question

We want to prove that: $$\aleph_2^{\aleph_0} = \aleph_2\aleph_1^{\aleph_0}$$ My idea was to approach this by doing a Schroder-Bernstein style argument and proving this by showing two inequalities, ...
5
votes
2answers
139 views

Reference request, self study

I'm looking for references (books/lecture notes) for : Cardinality without choice, Scott's trick; Cardinal arithmetic without choice. Any suggestions ? Thanks in advance.
5
votes
1answer
871 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
2answers
3k views

An infinite subset of a countable set is countable

In my book, it proves that an infinite subset of a coutnable set is countable. But not all the details are filled in, and I've tried to fill in all the details below. Could someone tell me if what I ...
5
votes
2answers
82 views

Why can't you count up to aleph null?

Recently I learned about the infinite cardinal $\aleph_0$, and stumbled upon a seeming contradiction. Here are my assumptions based on what I learned: $\aleph_0$ is the cardinality of the natural ...