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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43
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11answers
3k views

Refuting the Anti-Cantor Cranks

I occasionally have the opportunity to argue with anti-Cantor cranks, people who for some reason or the other attack the validity of Cantor's diagonalization proof of the uncountability of the real ...
1
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1answer
32 views

Let $R$ be an infinite comutative ring with unity, $M,N$ be $R$-modules, $f:M \to N$ be a surjective module homomorphism; then $|M|=|N ||\ker f|$?

Let $R$ be an infinite commutative ring with unity, $M,N$ be modules over $R$, let $f:M \to N$ be a surjective module homomorphism; then is it true that $|M|=|N || \ker f|$ ($M,N$ are not ...
4
votes
2answers
123 views

Counting subsets of different sizes of a set

Let $A$ be a non-empty set, and $\mathcal{P}^*(A)$ denote the power set of $A$ excluding empty set. There is a natural equivalence relation on $\mathcal{P}^*(A)$: for $S_1,S_2\in \mathcal{P}^*(A)$...
10
votes
0answers
110 views

Category-theoretic properties of cardinals

Let $\kappa$ be a cardinal, let $\mathbf{H}_\kappa$ be the set of hereditarily $\kappa$-small sets, and let $\mathbf{Set}_{< \kappa}$ be the full subcategory of $\mathbf{Set}$ corresponding to $\...
6
votes
2answers
203 views

Cardinality of the set of all infinite monotonically decreasing sequences of naturals

Find the cardinality of the set of all infinite monotonically decreasing sequences of naturals. I think it's $\aleph_0$. I marked this set in $A$, and said that $\forall n\in\Bbb N \ (n,n,n,...)\in ...
1
vote
1answer
32 views

Cantor's diagonal argument: Prove that $|A|<|A^{\Bbb N}|$

Let $A_1\subseteq A_2\subseteq A_3\subseteq...$ be a raising series of sets such that $\forall n\in \Bbb N \ |A_n|\lt |A_{n+1}|$. We mark $A$ as $A=\bigcup_{n\in\Bbb N}A_n$. Prove that $|A|<|A^{\...
0
votes
2answers
36 views

The cardinality of all the infinite binary sequences that don't contain 010

Find the cardinality of all the infinite binary sequences that don't contain 010 I think it's $\aleph_0$. I marked the set all infinite binary sequences that don't contain 010 in A, and the set of ...
18
votes
1answer
3k views

The cardinality of a countable union of countable sets, without the axiom of choice

One of my homework questions was to prove, from the axioms of ZF only, that a countable union of countable sets does not have cardinality $\aleph_2$. My solution shows that it does not have ...
1
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2answers
33 views

Cardinality of subsets with finite intersections

Let $\ F_0 $ be a family of disjoint subsets of $ C$. $\ |C|= \aleph_0$. Prove that $\ (*) |F_0|\leq\aleph_0 $. This part was relatively simple, in the presence of choice an injection can be ...
0
votes
0answers
34 views

Prove equinumerosity between $2^\mathbb{N}$ and R total orderings [duplicate]

T=$\left\{R\vert R \text{ is a total order over } \mathbb{N}\right\}$ Prove that T and $2^\mathbb{N}$ are equinumerous.
1
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0answers
33 views

For each of the following sets, determine its cardinality (ω, 2ω, or something else) and prove that your answer is correct

(a) A1 = {f ∈ (ω → ω) : ∀n,m ∈ ω (n < m ⇒ f(n) < f(m))}. (b) A2 = {f ∈ (ω → ω) : ∃n ∈ ω∀m ∈ ω f(m) ≤ n}. (c) A3 = {f ∈ (ω → ω) : ∃n ∈ ω∀m ∈ ω (n ≤ m ⇒ f(n) = f(m))}. a) A1 = {f ∈ (ω → ω) : ∀n,...
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 ...
1
vote
1answer
53 views

Proving the Cardinality of a set in R

Let $\ A\subset R $ have the following characteristic: For all $\ a,b \in A$ , $\ \frac{a+b}{2} \notin A$. Prove that there exists a maximal set A. Prove its cardinality is $\ \aleph $. The first ...
2
votes
2answers
54 views

A 'bad' definition for the cardinality of a set

My set theory notes state that the following is a 'bad' definition for the cardinality of a set $x:$ $|x|=\{y:y\approx x\}$ $(y\approx x\ \text{iff} \ \exists\ \text{a bijection}\ f:x\rightarrow y )...
0
votes
2answers
30 views

Paths in a connected subset of $\mathbb R^n$

Prove that every connected subset $X$ of $\mathbb R^n$ with more than one point has the continuum cardinality. In order to use Baire Lemma, I would like to demonstrate that there is a compact (or ...
2
votes
2answers
70 views

Existence of model of ZF in which every uncountable cardinal is the cardinality of some power set?

Does there exist a model of ZF in which any uncountable cardinal number is equal to the cardinality of the power set of some set ? ( In ZFC it is not possible as is shown by the answers to this ...
2
votes
1answer
43 views

Properties of the power set of $A$

Let $A$ be any set . Let $\wp(A)$ be the power set of $A$. Then which of the following are true 1) $\wp(A) = \emptyset$ for some $A$ 2) $\wp(A) $ is a finite set for some $A$ 3) $\wp(A)$ is a ...
1
vote
1answer
36 views

Convergence of real numbers along an ultrafilter

Suppose we have an uncountable set $I$ and a non-principal ultrafilter $U$ on it. I am interested whether it is possible to conclude that if cardinality of $I$ is big enough, then every tuple $(x_i)_{...
1
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1answer
26 views

Finding number of subsets of set S that have r elements in common with set T

I've been going crazy trying to solve this. The question asks For some $0 \le r \le k \le n$, how many subsets of {1...n} have r elements in common with the set {1..k}. Describe two sets S and T such ...
1
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2answers
71 views

$\aleph_1$ and $\omega_1$, what are they?

Sorry for my ignorant question but.. I understand that some sources says that $\aleph_1$ is the cardinality of the real numbers (ℝ) because In set theory $$\mathfrak{c} = 2^{\aleph_0} $$ and the ...
0
votes
2answers
40 views

Listing real numbers as countable like listing rational numbers [closed]

like proving the set of positive rational numbers are countable, where we list the rationals as the following list, why can't we represent real numbers like the same? If positive Rational numbers (p/...
1
vote
1answer
25 views

Determine the cardinality of $\{B\subseteq A \colon \vert B \vert \leq \kappa \}$

Let $A$ be a set. $\kappa$ a cardinal and assume that $\omega \leq \kappa \leq \vert A \vert \leq 2^{\kappa}$. Determine the cardinality of $C \colon=\{B\subseteq A \colon \vert B \vert \leq \kappa ...
2
votes
1answer
40 views

Are there any constructive axioms which disprove the continuum hypothesis?

I understand that the Continuum hypothesis is independent of ZFC, so that we may comfortably add either the continuum hypothesis or its negation to ZFC without creating any paradoxes (unless ZFC had ...
2
votes
1answer
32 views

How can I remember whether finite or countable cartesian product of countable set is countable

I always forget this result Is cartesian product of countable set countable under finite or countable cartesian products? Is there a good way to remember this? Like a proof sketch where the ...
1
vote
3answers
139 views

Good introduction to cardinals?

is there a good text book to cardinals? I am more interested in how cardinal works than cardinality. Because it seems in undergrad they cut off at proof of $\mathbb{R}$ is uncountable and does not go ...
3
votes
0answers
151 views

$\mathfrak b \leq \mathfrak r$. Is this proof correct?

I am trying to prove that $\mathfrak b \leq \mathfrak r$ where $\mathfrak r$ is the minimal cardinality of a reaping family of sets. A family $\mathcal R \subseteq [\omega]^\omega$ of sets $\mathcal ...
0
votes
2answers
45 views

Easy proof the set of finite Set in countable is countable [duplicate]

Suppose I know a result that the set of finite sets in $\mathbb{N}$ is countable. Is there a very quick way to show that the set of finite sets in any $X$ countable is countable? Idea...two sets ...
21
votes
2answers
2k views

Cardinality of a Hamel basis of $\ell_1(\mathbb{R})$

What is the cardinality of a Hamel basis of $\ell_1(\mathbb R)$? Is it deducible in ZFC that it is seemingly continuum? Does it follow from this that each Banach space of density $\leqslant 2^{\...
2
votes
0answers
26 views

If $A = \coprod_{n \in \mathbb{N}} A_n$ is uncountable, then there exists an uncountable $A_n$

Claim: If $A = \coprod_{n \in \mathbb{N}} A_n$ is uncountable, then there exists an uncountable $A_n$ $\coprod$ is the disjoint union of disjoint sets $A_n \subset A, \forall n \in \mathbb{N}$ Is ...
1
vote
1answer
63 views

Bijection from $\mathcal{P}(\mathbb{N})$ to $(0,1)$

How can we construct a bijection from $\mathcal{P}(\mathbb{N})$ to $(0,1)$? Here is what I know: $\mathcal{P}(\mathbb{N}) = \{A | A \text{ is a subset of } \mathbb{N}\}$ Both $\mathcal{P}(\...
4
votes
1answer
424 views

Largest infinite cardinal used in a proof

I've heard before that Knuth holds the record for the largest constant used in a mathematical proof. I was wondering what is the largest cardinal ever explicitly considered in set theory. I presume ...
0
votes
1answer
32 views

If $\lambda<|A|$, there exists $B \subset A$ such that $|B|=\lambda$

I've been thinking about the following claim: Let $A$ be a set and $|A|$ his cardinality. For every cardinal $\lambda$ with $\lambda<|A|$, there exists $B \subset A$ such that $|B|=\lambda$. ...
0
votes
3answers
35 views

Prove equal cardinality between two sets?

I'm preparing for a discrete math course in September and I'm trying to study on my own this summer. I've run into a bit of trouble with a practice problem I found online and can't really figure it ...
0
votes
3answers
32 views

If $(X, \mathcal{T})$ has a countable subbasis, then it has a countable basis

Given $(X, \mathcal{T})$ a topological space. Let $\mathcal{S}$ be a subbasis on $(X, \mathcal{T})$ Claim: If $\mathcal{S}$ is countable, then $\mathcal{T}$ has a countable basis $\mathcal{B}$ ...
1
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2answers
63 views

A specific example of transfinite induction

I am trying to understand better how transfinite induction can be applied in different concrete problems. Here is an example that seems relevant, but I am stuck on it. Consider a couple of points $p, ...
2
votes
1answer
59 views

Proving an existence of a cardinal when making assumptions on exponentiation

Let's assume $2^{\aleph_3}=\aleph_4$ and $\left(\aleph_{\omega_1}\right)^{\aleph_1}\neq\left(\aleph_{\omega_1}\right)^{\aleph_2}$. Prove that $$\exists_{\alpha\in Lim}\left( \left(\aleph_{\alpha}\...
2
votes
1answer
81 views

Well-orderings of $\mathbb R$ without Choice

The question is about well-ordering $\mathbb R$ in ZF. Without the Axiom of Choice (AC) there exists a set that is not well-ordered. This could occur two ways: a) there are models of ZF in which $\...
0
votes
0answers
36 views

Which is finer, co-countable topology or usual topology on $\mathbb{R}$?

We know that the usual topology is finer than co-finite topology on $\mathbb{R}$ How to show the usual topology is finer than co-finite topology on $\mathbb{R}$ And co-countable topology is (in ...
-1
votes
0answers
26 views

Partitioning an infinite set into sets of Aleph-Null [duplicate]

I was tasked with solving the following problem: Let $X$ be an infinite set. Prove using Zorn's Lemma that it is partition-able into subsets of cardinality $\aleph_0$. My initial thought was to try ...
2
votes
1answer
54 views

A nontechnical way to comprehend $\aleph_2$

This is possibly a dumb question, but I do not know where to look for an answer. Without getting technical, one can show why $card(\mathbb{N}) = card(\mathbb{Q}).$ (Typically by showing how the two ...
1
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0answers
43 views

Show $\mathbb{N}^{\{0,1\}}$ is uncountable with a hint

Let $\mathbb{N}^{\{0,1\}} :=\{f: \mathbb{N} \to \{0,1\}\}$ is uncountable I have never heard of the table approach, and all the proofs say uncountability of $\mathcal {P}(\mathbb{N})$ I have seen so ...
0
votes
2answers
36 views

Cardinal Numbers and powers related to them

Cardinal numbers are used to express sizes of sets. It $c$ is a cardinal number then $2^c$ is also a cardinal number which represents size of the set of all subsets of a set whose cardinality is $c$....
1
vote
1answer
29 views

Cardinality of infinite words from finite alphabet is the same as that of $\mathbb{R}$

Let $[n]$ denote the set $\{0, 1,2, \ldots, n\}$. I want to show that $[n]^{\mathbb{N}} \simeq \mathbb{R}$. I am aware you could do this with base $n$-ary expansions of reals, but that seems a bit ...
0
votes
2answers
46 views

Prove that $\mathbb{|Q| = |Q\times Q|}$

I have this problem: Prove that $\mathbb{|Q| = |Q\times Q|}$ I know that $\mathbb Q$ is countably infinite. But then how can I prove that $\mathbb{|Q\times Q|}$ is countably infinite? Thanks ...
0
votes
3answers
67 views

Which sets have cardinal number $\aleph_{0}$ or $\mathfrak{c}$?

(a) $[1,3)$, $\mathfrak{c}$ (b) $Z$, $\aleph_{0}$ (c)$R \times R$, (d) $R \cap Z$, (e) $\{ 2^{-k} : k \in \mathbb{N} \}$ I understand that aleph null means that it is infinite and that c means ...
1
vote
2answers
62 views

$|X|=|X\cup\{a\}|?$

Let $X$ be an infinite set and $a\notin X$. Prove $$|X|=|X\cup\{a\}|$$ This is so intuitively obvious but upon inspection it appears quite non-obvious. How might one prove this? Do I need the axiom ...
1
vote
2answers
111 views

Neighborhood chain in $\mathbb{R}^\mathbb{R}$

Edited after SamM's comment: Consider the topological space $\mathbb{R}^\mathbb{R}$, with the usual topology. Pick a point $x \in \mathbb{R}^\mathbb{R}$ and a neighborhood $V = V_0$ of $x$. I wish to ...
0
votes
1answer
11 views

Cardinal Inequality without using Choice

Let $k$ be an infinite cardinal (i.e. not in bijection with any natural number), show that $\mathbb{N} \leq 2^{2^k}$ This is very easy with choice, without it I don't even know where to start.
2
votes
2answers
76 views

Can the extended real number $+\infty$ be compared to transfinite numbers such as $\aleph_0$?

If not, why not? If so, is ∞ greater than or less than $\aleph_0$? Edit: the discussion in comments (including comments on a deleted answer) have made me think that the best way to put the issue is ...
0
votes
2answers
74 views

Does $1-\frac{1}{2^\omega}$ equal 1? What about $1-\frac{1}{2^{\aleph_0}}$?

(Correct if I'm wrong on any of this.) Recently, I've been learning about transfinite ordinals and cardinals. For some cases, I understand the difference between ordinals and cardinals, for instance ...