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 views

Number of equivalence classes based on a relation regarding a non-principal ultrafilter

We have an equivalence relation on $\mathbb{N}^\mathbb{N}$ given by $$f\equiv g \iff \{n\in\mathbb{N}: f(n)=g(n)\}\in\mathbb{U},$$ where $\mathbb{U}$ is a non-principal ultrafilter on $\mathbb{N}$. ...
0
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2answers
25 views

Finding an example of a bijection from $\Bbb N$ to $E^+$.

Give an example of a bijection $h$ from $\Bbb N$ to $E^+$ such that $h(1) = 16, h(2) =12, \text{ and } h(3) = 2. $ $\Bbb N = \text{ natural numbers }$ , $E^+= \text{ positive even integers. }$ So ...
8
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2answers
669 views

Is the set of aleph numbers countable?

If I write the set of aleph numbers in this way $\{\aleph_0, \aleph_1, \aleph_2, \aleph_3, \dots\}$ it seems obvious to me that this set is countable, because aleph numbers have integer coefficients. ...
2
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1answer
79 views

Without AC is there a relationship between $\beth$ and $\aleph$ numbers?

Assuming AC we know that all $\beth_\alpha$'s will be $\aleph_\beta$ for some $\beta$ since they can be well ordered. Can anything interesting be said about their relationship without AC? Is it ...
1
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1answer
174 views

Enderton's Elements of Set Theory Scott's Trick Exercise (page 207 problem 31)

31. Define kard $A$ to be the collection of all sets $B$ such that (i) $A$ is equinumerous to $B$, and (ii) nothing of rank less than rank $B$ is equinumerous to $B$. (a) Show that kard $A$ is a ...
0
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3answers
32 views

How to show that any separable space is CCC

I thought I had the proof of this in my head, but it doesn't sound right on paper. Can someone see if my argument could be improved. Let $(X,\tau)$ be a topological space that is separable, then it ...
5
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3answers
212 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 ...
1
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1answer
56 views

Show that a set with an uncountable subset is itself uncountable.

Let $A = P \cup Q$, where $P, Q$ are disjoint [1] and $P \ne \emptyset$ is countable and $Q \ne \emptyset$ is uncountable. Then $Q \subset A$ [2]. Show that $A$ is uncountable. Proof (by ...
1
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1answer
135 views

Did Cantor both prove and disprove the Continuum hypothesis?

I have been listening to the podcasts of A Brief History of Mathematics on BBC Radio 4. In the episode on Georg Cantor the narrator, Prof. Marcus du Sautoy, says that one day Cantor proved that there ...
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1answer
42 views

If a set $A$ is uncountable , and a set $B$ is countable then $A \times B$ is uncountable.

I prove it by contradiction. Let $A \times B$ is countable. It means we can list down the all the ordered pairs of $A \times B$. So if ordered pairs of the form $(a,b)$ are countable (where $a \in A$ ...
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3answers
51 views

Injective function between $\kappa^{\omega}$ and $[\kappa]^{\leqslant \omega}$

Is there an injective function $\varphi :\kappa^{\omega} \rightarrow [\kappa]^{\leqslant \omega}=\{ A\subset \kappa :|A|\leqslant \omega\}$ such that $\varphi (\alpha) \backslash \varphi (\beta)$ and $...
2
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1answer
23 views

Cardinality of a borelian

My advisor told me that a Borel set can only be finite, countably infinite or having the cardinality of the continuum (obviously we are not assuming Continuum Hypothesis). I think he mentioned "...
2
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1answer
44 views

$A \subseteq B \subseteq C ; A' \subseteq C' ; |A|=|A'| , |C|=|C'|$ ; then $\exists B' $ s.t. $A' \subseteq B' \subseteq C' $ , $|B|=|B'|$?

Let $X$ be a non-empty set and $A,B,C,A',C' \in \mathcal P(X)$ be such that $A \subseteq B \subseteq C ; A' \subseteq C'$ and $|A|=|A'| , |C|=|C'|$ ; then is it true that $\exists B' \in \mathcal P(...
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2answers
48 views

Lowering the cardinality of a set?

Given a set X with a certain cardinality, there are explicit constructions for getting a set with the "next bigger" cardinality, e.g. constructing the power set. Does some analogous construction ...
1
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1answer
50 views

prove sets cardinality inequality

I need to prove that if $$ A , B $$ are infinite sets and it holds that : $$ |A| > |B| $$ then: $$ |A \backslash B| = |A| $$ I guess I just don't what can I say about the cardinality of |A\B| ...
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1answer
32 views

Cardinality question on set of symbols [closed]

Few moments ago I asked myself a question, that I not positive if, in fact, is well defined. Let $\mathbb{R}$ be the set of real numbers. Define $S$ to be a set of symbols, as follows: Let $x$ be ...
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1answer
170 views

Does there exist a bijection from $[0,1]$ to $\mathbb R$?

We can find a bijection from $(0,1)$ to $\mathbb R$. For example, we can use $f(x)=\frac{2x-1}{1+|2x-1|}$ composed of parts of two hyperbolas, see the graph here. Or we could appropriately scale the ...
1
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1answer
31 views

Proving $x\preceq y \implies \bar{\bar{x}} \leq \bar{\bar{y}}$ in cardinal arithmetic

Let $\bar{\bar{x}}$ denote the cardinal of $x$ and $\approx$ denote bijective equivalence. Assume $x\preceq y$. By definition $\exists z (z \subseteq y \land x \approx z)$. Now from something I've ...
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4answers
190 views

Proving a set is Uncountable or Countable Using Cantor's Diagonalization Proof Method

I understand the idea that some infinities are "bigger" than other infinities. The example I understand is that all real numbers between 0 and 1 would not be able to "fit" on an infinite list. I have ...
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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
39 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
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2answers
132 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 $S_1,S_2\in \mathcal{P}^*(A)$...
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0answers
116 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 $\...
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2answers
205 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
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1answer
33 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
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2answers
38 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
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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 ...
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2answers
34 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 ...
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0answers
36 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.
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0answers
34 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,...
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2answers
535 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 ...
2
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1answer
54 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 ...
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2answers
55 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 )...
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2answers
31 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 ...
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2answers
76 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
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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
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1answer
38 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)_{...
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1answer
28 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 ...
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2answers
73 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 ...
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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/...
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1answer
26 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
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1answer
41 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
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1answer
33 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 ...
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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 ...
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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 ...
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2answers
46 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
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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
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0answers
27 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
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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
426 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 ...