Tagged Questions

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.

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1
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1answer
21 views

Cardinality of the power set $\mathcal P\left(S\right),$ where $S$ is a set of $15$ elements?

What is the cardinality of the power set $\mathcal P\left(S\right)$ where $S$ is a set of $15$ elements? I think the power set is a set of all the subsets of a given set or $2^n$. So would the ...
0
votes
0answers
21 views

Prove that for a subset $A$ of $\mathbb{R}$, if $|A| = \omega$ then $|\mathbb{R} - A| = 2^\omega$ [duplicate]

I'm just starting with fundamental set theory and am trying to solve the following problem: Prove that for a subset $A$ of $\mathbb{R}$, if $|A| = \omega$ then $|\mathbb{R} - A| = 2^\omega$. I think ...
6
votes
2answers
513 views

Could someone explain aleph numbers?

I am having trouble understanding aleph numbers. I understand $\aleph_0$ is a countable infinity, but after that, I'm lost. What are $\aleph_1,\aleph_2,\aleph_3$, etc. to $\aleph_n$? Is there an ...
2
votes
2answers
58 views

Cardinality of the set of all numbers that modern math can define?

I have recently learned that the algebraic numbers are countably infinite, and that very few transcendental numbers are known. Are enough transcendentals known to make up an uncountable set, or is ...
1
vote
1answer
34 views

Prove that $\omega_{\alpha+1}$ is regular

Here, under the section 'Regular and Singular Cardinal', there is this sentence 'Assuming the Axiom of Choice, $\omega_{\alpha+1}$ is regular for each $\alpha$' . May I know how to prove this? Also, ...
0
votes
1answer
28 views

Haudorff Formula Set Theory

For every $\alpha$ and every $\beta$, $$\aleph_{\alpha+1}^{\aleph_{\beta}}=\aleph_{\alpha}^{\aleph_{\beta}} \cdot \aleph_{\alpha+1}$$ Proof: If $\beta \geq \alpha+1$, then ...
3
votes
3answers
77 views

Is there an infinite countable $\sigma$-algebra on an uncountable set

Let $\Omega$ be a set. If $\Omega$ is finite, then any $\sigma$-algebra on $\Omega$ is finite. If $\Omega$ is infinite and countable, a $\sigma$-algebra on $\Omega$ cannot be infinite and ...
1
vote
1answer
23 views

Cardinality for Kleene star and infinite Cartesian products.

Let $X$ be a finite set with at least 2 elements. Then the set of all finite-length "strings", $$X^* = \bigcup_{L \in \mathbb{Z}^+} \prod_{i=1}^L X_i = \{ (x_1, \ldots, x_L) : L \in \mathbb{Z}^+ ...
0
votes
1answer
60 views

Infinite Cardinal Addition Without the Axiom of Choice

In the book 'Introduction to Set Theory' by Hrbacek and Jech, cardinal addition is defined as $$\sum_{i \in I}{\kappa_i}=\left|\bigcup_{i \in I}{A_i} \right|$$ where $|A_i|=k_i$ for all $i \in I$ ...
3
votes
3answers
60 views

$|A\times B|= \text{max}(|A|,|B|)$ for infinite sets

I am fairly sure, given examples $\Bbb{R}\times \Bbb{R},\Bbb{R}\times \Bbb{Q},\Bbb{Q}\times \Bbb{Q} $, that this is correct, but do not know how to prove it. In my cited examples the proof has ...
0
votes
1answer
34 views

Mapping between set cardinalities

Please help me prove the following equalities between set cardinalities by explicitly showing an appropriate mapping: $$\left | (0,1) \right |= \left | (1,+\infty ) \right |$$
-1
votes
0answers
25 views

Mapping between two set equalities

Please help me prove the following equalities between set cardinalities by explicitly showing an appropriate mapping: $$\left | (0,+\infty ) \right |= \left | \mathbb{R} \right |$$
1
vote
1answer
44 views

Comparing cardinalities

Why these two sets are equinumerous? $$[0,1]^\Bbb N\text{ and }\Bbb Q^\Bbb N$$ Here is my reason: The set of rational numbers $\Bbb Q$ is countably infinite. However, $[0, 1]$ is not countable and ...
0
votes
0answers
43 views

Cardinal Arithmetic Example Wikipedia

Hello I am studying cardinal arithmetic, and found out that I found that $\mathfrak{c}^{\aleph_0} = (2^{\aleph_0})^{\aleph_0} = 2^{\aleph_0 \aleph_0} = 2^{\aleph_0} = \mathfrak{c} $. However I found ...
0
votes
0answers
22 views

Prove that the set is countable [duplicate]

Question: We call a real number $x$ $algebraic$ if $x$ is the root of a polynomial equation $c_{0}+c_{1}x+...+ c_{n}x^{n}$ where all $c_{i}$'s are integers. For example $\pm \sqrt{3}$ are algebraic ...
3
votes
1answer
46 views

Is it possible to define a map from $\aleph_0$?

In an assignment question I am asked to show that the cardinality of the set of all functions from $\mathbb{N}$ to $\mathbb{N}$ is equal to $2^{\aleph_0}$. To proceed with my proof I am trying to ...
0
votes
2answers
57 views

Show that $\aleph_{\alpha} +\aleph_{\alpha} = \aleph_{\alpha}$ and $n \cdot \aleph_{\alpha} = \aleph_{\alpha}$

a) Give a direct proof of $\aleph_{\alpha} +\aleph_{\alpha} = \aleph_{\alpha}$ by expressing $\omega_{\alpha}$ as a disjoint union of two sets of cardinality $\aleph_{\alpha}$. b) Give a direct proof ...
0
votes
0answers
18 views

Logarithms of Cardinals [duplicate]

Given any infinite cardinal $\lambda\neq\omega$, is it the case that there's a cardinal $\kappa$ such that $2^{\kappa}=\lambda$? Does this depend on whether the Continuum Hypothesis is true? Clearly, ...
6
votes
2answers
102 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 ...
3
votes
1answer
63 views

If $2^{\aleph_0}$ is weakly inaccessible, can every cardinal $\kappa$ in the interval $[\aleph_0,2^{\aleph_0})$ satisfy $2^\kappa = 2^{\aleph_0}$?

Question. Is the following consistent with ZFC? $2^{\aleph_0}$ is weakly inaccessible Every cardinal $\kappa$ in the interval $[\aleph_0,2^{\aleph_0})$ satisfies $2^\kappa = ...
3
votes
2answers
63 views

$0^\sharp$ and the regularity of $\aleph_\omega$

I'm sure I'm missing something trivial, and the most likely of it is that I'm simply wrong on my understanding of the constructible universe $L$, or maybe one of the Wikipedia entries I'm about to ...
4
votes
3answers
136 views

An easy to understand definition of $\omega_1$?

I have two things I'm not sure in 100% about them. The first, is $\omega_1$. I have a little "feeling" of it, but if I'll be asked to define it - I don't know where to begin from. Perhaps it is ...
4
votes
0answers
37 views

Can every cardinal number between $\kappa^+$ and $2^\kappa$ be realized in this way?

(Assume ZFC for the entire question. By a tree, I mean a tree in the sense of set theory. I write $h(T)$ for the height of a tree $T$, and $h(f)$ for the height of an element $f \in T$.) Definition ...
0
votes
1answer
35 views

dimension of direct products

Suppose $\{V_i\}_{i\in I}$ is a family of $k$ vector spaces. Is it possible to calculate $\dim\oplus_{i\in I} V_i$ and $\dim\prod_{i\in I}V_i$?
2
votes
1answer
41 views

Cardinality: is it true that $|X^\mathcal{inj}| \leq 2^{|X|}$?

Definition. Whenever $X$ is a set, write $X^\mathcal{inj}$ for the collection of all injections $f$ such that: $f$ has codomain $X$ There exists an ordinal $\alpha$ such that $f$ has domain ...
1
vote
0answers
38 views

Is it true that $\dim(X) \leq \dim(X^{\ast})$ for every infinite dimentional banach space $X$?

So given an arbitrary infinite dimensional Banach space $X$ can we deduce that it's dimension $\dim(X)$ (the cardinality of one of it's Hamel bases) is less or equal of the dimension $\dim(X^{\ast})$ ...
1
vote
1answer
71 views

Why coefficients of Fourier series are countable, though the initial periodic function is described with an uncountable set of points

Coefficients in the Fourier series for any periodic square-integrable function $f(x)$ form a countable (though infinite) set, i.e., they have cardinality $\aleph_0$. As far as Fourier exponents form a ...
2
votes
1answer
69 views

How to force p<b?

Two cardinal characteristics (cardinals between $\aleph_1$ and $\mathfrak{c}$ are: $\mathfrak{b}$, the least size of an unbounded family in $\omega^{\omega}$ ordered under eventual domination ...
1
vote
2answers
54 views

When is an infinite set larger than another infinite set?

Somewhat of a basic question that I've been pondering about, suppose we have 2 finite sets $A,B$, arbitrary sets with arbitrary elements that we know nothing about, except that they are both finite. ...
0
votes
1answer
16 views

Building a function with codomain equal to a given set of reals.

I was discussing with friends the astounding fact that $\mathbb R$ and the set of real continuous functions were equipotent. I asked for a proof that $\mathbb R$ and $\mathbb R ^{\mathbb R}$ are not ...
2
votes
3answers
51 views

Does there exist this type of sequence of subsets of $\mathbb{R}$?

Let $A_n$ be a sequence of subsets of $\mathbb{R}$ with the following properties. $A_n$ is unbounded for all $n$ The union of all $A_n$ is $\mathbb{R}$ No two $A_n$ share elements For all $n$, given ...
1
vote
1answer
14 views

Would 3 to the n power where n is an element of Z be countably infinite?

I'm just learning about finite, countably infinite, and uncountable sets. My question is, which of the three categories would this fall into: {3^n|n ϵ Z} I first thought that it would uncountable, ...
-1
votes
2answers
18 views

One-to-one correspondence of a set within a set

I need to find a one-to-one correspondence between each of the following pairs of sets: (a) {x, y, {a, b, c}} and {14, -3, t} (b) 2Z and 17Z For problem a, I have no idea if the inner set counts as ...
0
votes
1answer
28 views

Prove cardinal arithmetic (exponentiation)

Suppose $|K|=\kappa, |L|=\lambda, |M|=\mu$ and $L \cap M=\emptyset$. Prove that $$(\kappa^{\lambda})^{\mu}=\kappa^{\lambda \cdot \mu}$$ My attempt: Suppose $F : K^{ L \times M} \rightarrow (K^L)^M$. ...
0
votes
1answer
44 views

Dedekind finite set and a special well ordered set

In ZFC, Dedekind finite set and finite set are same things. So I have a set say A(which is equal to N in ZFC) all Dedekind finite set are equivalent to proper subsets of A and A is well ordered set. ...
0
votes
0answers
24 views

Cardinality of a set of injections [duplicate]

Let $A$ be the set of all injections $f: \mathbb{Z}_+ \rightarrow \mathbb{Z}_+$ What can we say about the cardinality of $A$ with respect to the cardinalities of $\mathbb{Z}_+$ and ...
0
votes
1answer
20 views

For every ordinal $\alpha$, there is a cardinal number greater then $\alpha$.

I am trying to prove that for every ordinal $\alpha$, there is a cardinal number greater then $\alpha$. Proof: $\alpha$ is a (well ordered) set. Take the set $P(\alpha)$. Assume that we know that for ...
1
vote
0answers
33 views

Decomposition of an infinite set into pairwise disjoint subsets which exhaust set does not affect cardinality

Show that if $X$ is an infinite set and $A$ is subset of the power set of $X$ containing only finite pairwise disjoint sets such that the union of all elements of $A$ is $X$, then cardinalities of $X$ ...
2
votes
1answer
51 views

Need some help with this Cardinality/sets question.

I've got this problem about sets, and cardinality. I don't really understand it other than cardinality is the number of elements within each set, I don't understand a lot of the signs used within the ...
1
vote
1answer
47 views

Cardinal exponentiation formula

Assume GCH and let $k,m$ be infinite cardinals. I would like to show that $k^m = \max \{ k,2^m \}$. We of course have $k=\beth_a$ and $m=\beth_b$ for ordinals $a,b$. If $a$ is a successor ordinal, ...
-2
votes
2answers
58 views

What is aleph null times aleph one?

Could you shed some light on this? I am guessing it is aleph one, since one cannot pair every element of naturals with its subsets.
2
votes
1answer
22 views

Prove there's $\left|A-B\right| = \aleph$.

Let A a set such that $\left|A\right| \ge \aleph$. Prove there's a $B\subseteq A$ such that $\left|B\right|\ge \aleph$ and $\left|A-B\right| = \aleph$. Lets assume there's a $B\subseteq A$ such ...
1
vote
2answers
49 views

Cardinality of all injective functions from $\mathbb{N}$ to $\mathbb{R}$.

What is the cardianlity of: $$ A = \left\{ f:\mathbb{N}\to\mathbb{R} : \text{f is injective} \right\} $$ Trying to prove it using Cantor–Bernstein–Schroeder theorem, I have the obvious side: $$A ...
1
vote
1answer
41 views

Show $\left|B^A\right| \cdot \left|B^A\right| = \left|B^A\right|$

Let $A$, an infinite set such that $\left|A\right|\cdot \left|A\right| = \left|A\right|$ and Let $B$, an arbitrary set. Show $\left|B^A\right| \cdot \left|B^A\right| = \left|B^A\right|$ I'd be ...
1
vote
2answers
83 views

Sizes of infinity

I was just thinking about infinity (as you do) and thought the following. "There are infinitely many reals in the interval $x\in[0,1]$ and an 'equal number of reals' $x\in[1,2]$, so there are 'double ...
2
votes
2answers
31 views

Show $\left|B-A\right| = \left|B'-A'\right|$

Let $A,A',B,B'$ such that: $A\subseteq B$, $A'\subseteq B'$, $\left|B\right|=\left|B'\right| \gt \aleph_0$, $\left|A\right|=\left|A'\right| = \aleph_0$. Show that $\left|B-A\right| = ...
0
votes
0answers
87 views

Existence of a Banach space with algebraic dimension $\alpha \neq \aleph_0$

For any given cardinal number $\alpha \neq \aleph_0$, is there a Banach space $X$ with algebraic dimension $\alpha$?
4
votes
1answer
36 views

Is there a cardinal $\kappa$ with uncountable cofinality such that $\kappa<\kappa^{\aleph_0}$?

Is there a cardinal $\kappa$ with uncountable cofinality such that $\kappa<\kappa^{\aleph_0}$? The question is motivated by the observation that $\kappa< \kappa^{{\rm cf}\kappa}$ for any ...
1
vote
1answer
22 views

Why is the equality right? (Set-Theory)

Let $A, B$ finite sets, and let $f,g\in A\to B$. Also, Let the equivalence class: $$f \sim g \iff \exists h\in Eq(A,A). f=g\circ h $$ Claim: $$f\sim g \iff \forall b\in B. \left| \left\{ a\in A : ...
1
vote
2answers
24 views

What is the cardinality of $\left|C_s\right|$?

Let $$C_s = \left\{ f\in \mathbb{N}/S \to \mathbb{N} : \forall M\in \mathbb{N} / S. f(M)\in M \right\}$$ Where $S$ is an equivalence class. I need to prove $$\left|C_s\right| > \aleph_0 \implies ...