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)

8
votes
3answers
196 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 ...
14
votes
4answers
5k 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 ...
6
votes
1answer
226 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?
3
votes
3answers
119 views

How to understand $\operatorname{cf}(2^{\aleph_0}) > \aleph_0$

As a corollary of König's theorem, we have $\operatorname{cf}(2^{\aleph_0}) > \aleph_0$ . On the other hand, we have $\operatorname{cf}(\aleph_\omega) = \aleph_0$. Why the logic in the latter ...
3
votes
2answers
305 views

The cofinality of $\aleph_{\omega\cdot9+3}$

I am studying for a test and I was able to find the cofinality 3 of the 4 ones given, but am having a lot of trouble with the 4th. the 3 first ones are: ...
3
votes
1answer
83 views

The product of the finite non-zero ordinals, and the product of the finite non-zero cardinals.

I am trying to study for a test I have on Set Theory, and after being given the practice test, I am having some big problems with simple things. One question I am having a lot of problems with is: ...
2
votes
2answers
112 views

Prove that $S$ is countably infinite

Suppose we have the set $S\subset\mathbb{ N \times N}$ where $\mathbb N$ is the set of positive integers $\{1, 2,\ldots\}$ with the property: $S = \{\langle m, n \rangle\mid m \leq n\}$. Suppose ...
3
votes
3answers
290 views

Why the principle of counting does not match with our common sense

Principle of counting says that "the number of odd integers, which is the same as the number of even integers, is also the same as the number of integers overall." This does not match with my ...
4
votes
1answer
442 views

Counterexamples to the continuum hypothesis

Assume the continuum hypothesis is false, and add that as an axiom to ZF set theory. How many cardinalities are between the rationals and the reals in this case? Only one? Infinitely many? Countably ...
7
votes
2answers
424 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 $M$?
1
vote
1answer
119 views

Cardinality natural numbers

Is the cardinality of the natural numbers a natural number? $|\mathbb{N}| \in \mathbb{N}\text{ or } |\mathbb{N}| \notin \mathbb{N}$, that is the question.
0
votes
1answer
492 views

Infinite Dedekind Finite sets

I realized that i have used argument below many times before and I'm not sure if it is true. Let $A=\{n\in \omega|\Phi(n)\}$. Then $A\preceq \aleph_0$. (i)Suppose $A$ is dedekind-infinite and find ...
1
vote
4answers
141 views

Which kind product of non-zero number non-zero cardinal numbers yields zero?

Let $I$ be a non-empty set. $\kappa_i$ is non-zero cardinal number for all $i \in I$. If without AC, then $\prod_{i \in I}\kappa_i=0$ seems can be true(despite I still cannot believe it). But what ...
7
votes
1answer
546 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
1answer
214 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}) $ ...
0
votes
4answers
352 views

Comparing the cardinality of sets

An exercise is the following: Compare the cardinality of the following sets: The class of all real numbers $\mathbb{R} =: A$ The class of all polynomials $\mathbb{R}[X] =: B$ The class of all real ...
4
votes
1answer
79 views

A cardinal $\kappa$ such that $2^{\lambda}<\kappa$ for all $\lambda<\kappa$ is regular?

A cardinal $\kappa$ such that $2^{\lambda}<\kappa$ for all $\lambda<\kappa$ is regular? I would appreciate very much an answer
1
vote
2answers
351 views

Cardinality of Sets and Infinite Sets

The following are homework questions I would like assistance on. I will do what I can to work on these problems; any feedback is helpful. In the following problems, S is an infinite set (we do not ...
4
votes
4answers
673 views

Cardinality of Vitali sets: countably or uncountably infinite?

I am a bit confused about the cardinality of the Vitali sets. Just a quick background on what I gather about their construction so far: We divide the real interval $[0,1]$ into an uncountable number ...
0
votes
2answers
2k views

Definition of denumerable (countable) set

When we say that a set $S$ is denumerable, that is, there is a bijection $S \to \omega$, do we mean that there exists such a bijection or do we mean that we have one and are talking about a pair ...
2
votes
4answers
672 views

What's the definition of $\omega$?

This is a follow up on a comment to one of my previous questions. What's the definition of $\omega$? Are the following equivalent definition of $\omega$: $\omega$ is the initial ordinal of ...
1
vote
3answers
174 views

Question about models, cardinalities and collapsing

I have a (seeming) contradiction and I can't seem to figure out the (obvious) mistake in my reasoning. The following (related to my previous question): (1) $\omega$, defined to be the least infinite ...
9
votes
2answers
255 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 ...
2
votes
2answers
155 views

Possible inaccuracy in Wikipedia article about initial ordinals

I quote from the Wikipedia article: "So (assuming the axiom of choice) we identify $\omega_\alpha$ with $\aleph_\alpha$, except that the notation $\aleph_\alpha$ is used for writing cardinals, and ...
1
vote
3answers
186 views

cardinality problem

Let $\mathbb{R}^\mathbb{R}$ be the set of all functions $f: \mathbb{R} \to \mathbb{R}$ and $P(\mathbb{R})$ be the power set of $\mathbb{R}$. How to show that they have the same cardinality?
3
votes
4answers
239 views

Is this set bijective to R?

The set of all infinite sequences with integer entries? Obviously my set is at least as large as R, but is it larger?
4
votes
4answers
527 views
1
vote
2answers
190 views

What is the cardinality of a transcendence basis of $\mathbb{C}$ over $\mathbb{Q}$?

What is the cardinality of a transcendence basis of $\mathbb{C}$ over $\mathbb{Q}$? Is it that of the continuum? Proof?
0
votes
1answer
124 views

Contour Infinites and Vector Spaces

We usually define Hilbert or finite dimensional vector spaces, and even topologies or differential geometry on $\mathbb{R}^n$ , so I wonder what is the implication of doing that on some extended ...
5
votes
5answers
327 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 ...
1
vote
2answers
136 views

Sur- in- bijections and cardinality.

I think about surjection, injection and bijections from $A$ to $B$ as $\ge$, $\le$, and $=$ respectively in terms of cardinality. Is this correct? And extrapolating from that, are these theorems ...
8
votes
1answer
466 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
5
votes
1answer
218 views

Why is ${\aleph_\omega}^{\aleph_1} = {\aleph_\omega}^{\aleph_0} \cdot {2}^{\aleph_1}$? [duplicate]

I am supposed to prove that ${\aleph_\omega}^{\aleph_1} = {\aleph_\omega}^{\aleph_0} \cdot {2}^{\aleph_1}$ , but I really have no idea how to start or what to do. I thought I could use the following ...
5
votes
1answer
308 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? ...
4
votes
2answers
141 views

Is $2^{|\mathbb{N}|} = |\mathbb{R}|$?

Is $2^{|\mathbb{N}|} = |\mathbb{R}|$? If so, how? I was reading the Wiki page on the , and it says "Moreover, $\mathbb{R}$ has the same number of elements as the power set of $\mathbb{N}$", but I ...
3
votes
3answers
536 views

Understanding the proof for $ 2^{\aleph_0} > \aleph_0$

I'm trying to understand how $ 2^{\aleph_0} > \aleph_0 $. I was reading through this sketch of the proof, but don't quite understand how they show that $\mathrm{card}((0,1)) = ...
0
votes
2answers
363 views

Infinite union of sets, with cardinality of each is a unique cardinal number.

I have a proof for the following problem but I am not sure if it is valid or generalizes to infinite number of sets. Please give your suggestions or a better proof. $|pow(\mathbb{N})|$ is defined as ...
2
votes
1answer
193 views

Bijection from $(0,1]$ to $[0, \infty)^2$

Define a bijection from $(0,1]$ to $[0, \infty)^2$ Route to follow, A-) First define a bijection from $(0,1]$ to $(0,1]^2$ B-) Since there is a bijection from $(0,1]$ to $[0, \infty)$, namely $f(x) ...
1
vote
3answers
45 views

Is this definition true about comparisons of sets?

I'm trying to find the error in a proof that yields a contradictory result, and I'm beginning to think that one of the definitions I start with is incorrect or self-contradictory. Is the following ...
1
vote
1answer
190 views

Cardinality of two infinite sets and strictly bigger than relation

Suppose $A$ and $B$ are two infinite sets and cardinality of $B$ is strictly greater than cardinality of $A$. To prove this strictly gretater than relation between two sets, is it sufficient to only ...
6
votes
3answers
2k 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?
4
votes
3answers
345 views

Proving $\mathbb{R}$ is uncountable using Dedekind cuts?

I'm familiar with several proofs that the real numbers are uncountable (Cantor's initial proof, a proof by diagonalization, etc.). However, I've never seen a proof that the reals are uncountable that ...
7
votes
1answer
142 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 ...
3
votes
2answers
331 views

Cardinality of an infinite separable connected metric space is $2^{\aleph_0}$.

How to prove: Cardinality of an infinite separable connected metric space is $2^{\aleph_0}$. Thanks in advance!!
2
votes
2answers
153 views

Cardinal arithmetic and first uncountable cardinal

Cardinal arithmetic does not seem to open its way to the existence of $\aleph_1$ that is not $2^{\aleph_0}$, as any operation on $\aleph_0$ would lead to $\aleph_0$ or $2^{\aleph_0}$ and ...
5
votes
3answers
419 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 => |\mathbb R| = ...
4
votes
3answers
463 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 ...
6
votes
1answer
418 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
258 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.
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 ...