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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Showing $\prod_{n < \omega} n = 2^{\aleph_0}$ [duplicate]

I have to show that $\prod_{n < \omega} n = 2^{\aleph_0}$. I'm having trouble getting started. I know that $2^{\aleph_0}$ is the set of binary sequences, or the space of functions from $\mathbb{N}$ ...
3
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1answer
78 views

$\kappa$ ineffable $\Rightarrow$ $\kappa$ tree-property

Let $\kappa$ be an uncountable, regular cardinal. We call $\kappa$ ineffable iff for every sequence $(A_\xi \colon \xi < \kappa)$ of subsets $A_\xi \subseteq \xi$ there is a stationary subset $S ...
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3answers
190 views

The Free Set Lemma

The statement of the lemma is as follows: if $$f: \omega_1 \rightarrow \{x\ :\ x\ \textrm{is finite}\}$$ then there is an uncountable $S \subseteq \omega_1$ such that for all distinct $\alpha,\ \beta ...
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1answer
44 views

How to define an explicit bijection from P(N) to 2^N [closed]

How do I define an explicit bijection between the power set of N and $2^N$ with $2^N =\{f|f:N\to\{0,1\} \text{ is a function} \}$?
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2answers
25 views

How do I prove an equivalence of these two statements about Cantor's hypotheses?

How do I prove that the continuum hypotheses as stated by George Cantor (There are no sets with cardinality between the cardinality of the real and the cardinality of the rational numbers) is ...
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1answer
30 views

smallest cardinal greater than an infinite ordinal is a regular cardinal

let $\alpha$ be an infinite ordinal, and $\alpha^+$ be the smallest cardinal greater than $\alpha$. Show that $\kappa^+$ is a regular cardinal. This is for homework, but I'm not really sure where to ...
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0answers
63 views

$\mu$-clubs and stationary sets consisting of elements with cofinality $\mu$

Let $\mu < \kappa$ be infinite cardinals. A set $C$ is called a $\mu$-club in $\kappa$, if it is unbounded in $\kappa$ and contains all its limit points of cofinality $\mu$. Now let $T \subset S ...
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1answer
62 views

If $2^{\aleph_{\beta}}\geq \aleph_{\alpha}$, then $\aleph_{\alpha}^{\aleph_{\beta}}=2^{\aleph_{\beta}}$.

If $2^{\aleph_{\beta}}\geq \aleph_{\alpha}$, then $\aleph_{\alpha}^{\aleph_{\beta}}=2^{\aleph_{\beta}}$. Proof: Note that if $\beta \geq \alpha$, then we have ...
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1answer
50 views

Cardinality question on vector spaces

Suppose $F$ is a field and $J$ an infinite set. Is it then true that $\mathrm{card} \ J<\mathrm{card} \ F^J$? ($F^J$ the set of maps $J\rightarrow F$) I know that $\leqslant $, but is it true and ...
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2answers
96 views

The cardinality of a union of two sets

Assume that the cardinality of the union of two sets is continuum. How to prove that at least one of the sets has the cardinality of a continuum? I suppose that it's possible to cope with it, using ...
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1answer
31 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 ...
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2answers
590 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
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2answers
72 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 ...
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1answer
39 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, ...
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1answer
32 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 ...
4
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3answers
158 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 ...
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1answer
75 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}^+ ...
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1answer
79 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
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3answers
70 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 ...
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1answer
40 views

Mapping between set cardinalities

Please help me prove the following equalities between set cardinalities by explicitly showing an appropriate mapping: $$\left | (0,1) \right |= \left | (1,+\infty ) \right |$$
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1answer
60 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 ...
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1answer
68 views

Cardinal Arithmetic Example Wikipedia

Hello I am studying cardinal arithmetic, and found out that: $$\mathfrak{c}^{\aleph_0} = (2^{\aleph_0})^{\aleph_0} = 2^{\aleph_0 \aleph_0} = 2^{\aleph_0} = \mathfrak{c} $$ However I found this ...
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1answer
71 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 ...
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2answers
64 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 ...
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0answers
19 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, ...
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2answers
123 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
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1answer
75 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
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2answers
79 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
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3answers
162 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
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0answers
43 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 ...
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1answer
39 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 $\dim\prod_{i\in I}V_i$?
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1answer
43 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 ...
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0answers
42 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 $\dim(X^{\ast})$ ...
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1answer
105 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 ...
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1answer
83 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 ...
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2answers
67 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
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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 ...
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3answers
58 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 ...
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1answer
17 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, ...
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2answers
42 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: $\{x, y, \{a, b, c\}\}$ and $\{14, -3, t\}$ $2\mathbb Z$ and $17 \mathbb Z$ For problem a, I have no idea if ...
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1answer
102 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 $F : K^{ L \times M} \rightarrow (K^L)^M$. ...
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1answer
50 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. ...
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0answers
26 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
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1answer
26 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 ...
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0answers
46 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
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1answer
80 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 ...
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1answer
69 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, ...
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2answers
82 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
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1answer
25 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 ...
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2answers
106 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 ...