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 $\mathbb{N}$. ...
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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 ...
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 ...
8
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2answers
668 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. ...
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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 ...
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 ...
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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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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 ...
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 "...
0
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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 $...
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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| ...
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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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 ...
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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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 ...
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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 ...
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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^{\...
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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 ...
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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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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,...
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 ...
2
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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 ...
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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 ...
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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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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 ...
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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 ...
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 ...
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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}(\...
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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 $|B|=\lambda$. ...
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3answers
36 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 ...
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3answers
33 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 $\mathcal{B}$ ...
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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 ...
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2answers
66 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, ...
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1answer
62 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
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1answer
82 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 $\...
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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 ...
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1answer
56 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 ...
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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 ...
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2answers
38 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 $c$....
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1answer
30 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 ...
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3answers
75 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 ...
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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 ...