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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2
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1answer
56 views

Show that the union of two sets of the same cardinality has again the same cardinality.

Greetings great wise ones. Continuing my set-theoretic adventures, I have again stumbled upon a problem and need guidance. The original problem goes like this: Let $k_0 = \aleph_0$ and for any $n &...
0
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2answers
58 views

Unique infinite subsets of the integers

Edit: Great points on the comments. There is no unique set of unique infinite subsets of the integers. Is this a better question? What is the largest possible cardinality of a set which is a set of ...
27
votes
3answers
952 views

What is the largest set for which its set of self bijections is countable?

I recently came across a problem which required some knowledge about the self bijections of $\mathbb{N}$, and after looking up how to construct some different bijections I came across the result that ...
1
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0answers
44 views

Cardinality of Binary sets

Two questions I encountered in my last Set Theory HW. 1) Let T be a set of all Binary sequences that do not contain 2 consecutive zeros (ex. $100111010\notin T$). Let B be a set that contains all ...
4
votes
3answers
100 views

The cardinality of Indra's net?

This question has been asked before, but the title of the post was so general that it received no sufficient answer. What is the cardinality of the set of jewels and reflected jewels in Indra's Net? ...
7
votes
1answer
184 views

Can all Power Sets be Limit Cardinals?

Is it possible to create a model of ZFC, so that the cardinality of each power set is a limit cardinal (as opposed to GCH where they are always successor cardinals)? Take for example the following ...
11
votes
1answer
222 views

Cardinality of power sets decides all of cardinal arithmetic?

Assuming ZFC, is it possible to have two models which agree on the cardinality of all the power sets, but disagree on the cardinality of some other cardinal exponentiation (meaning that they agree on ...
0
votes
1answer
34 views

Defining a function that maps two sets

I am new to the topic of cardinality and I am trying to prove the following statement: "If $a$ is a natural number then $\mathbb{N} \setminus \{ a \}$ is denumerable. Here, $\mathbb{N} \setminus \{...
0
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2answers
40 views

Applying power set finite times

Is every infinite set $A$ smaller than a set of the form $\mathcal P (\mathcal P(\dots \mathcal P(\mathbb N)))$?
2
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1answer
29 views

cardinal arithmetic: prove that $k_2^m>=k_1^m$ if $k_2>=k_1$

I could use some help on this problem: Suppose $k_1$, $k_2$, $m$ are cardinals. Given that $ k_2 \geq k_1$ prove that $ k_2^m\geq k_1^m$ . I know that I need to find a one to one function $f$ from $...
2
votes
3answers
79 views

Is there a set $A$ such that $|\mathbb Z|<|A|<|\mathbb R|$ is undecidable?

CH guarantees that the statement $|\mathbb Z|<|A|<|\mathbb R|$ is false for all $A$, but since $\sf CH$ is undecidable it might still be possible that there exist a set $A$ for which the ...
2
votes
2answers
83 views

Can I assume the continum hypothesis in a proof

I am showing that the cantor ternary set has the same cardinality as $\mathbb{R}$ I want to use the fact that it is uncountably infinite and a subset of $\mathbb{R}$. ($|N| < |C| \leq \mathbb{R}$) ...
4
votes
4answers
45 views

A surjective map $S \to T$ implies $|S| \geq |T|$

Problem: Suppose that there is a function mapping $S$ onto $T$. Show that $\operatorname{Card}(S)\ge\operatorname{Card}(T)$ Issue: I can't seem to find a reason why this follows. If $S$ maps $T$...
2
votes
1answer
22 views

Possible cardinalities of the equivalence partitioning

Let $\sim$ denote a relation in $\mathbb{R}$ as follows: $x \sim y \iff d(x,y) \in \mathbb{Q} $ ($d(x,y)$ is the distance between $x$ and $y$) Determine the possible cardinalities of the equivalence ...
2
votes
0answers
31 views

Cardinality of this equivalence class

I'm looking at the following equivalence relation on $\mathbb{Z}$: $a \sim b$ if and only if there exist $n,m \in \mathbb{N}_{>0}$ so that $a^n = b^m$ I'm trying to determine what the cardinality ...
1
vote
2answers
63 views

Cardinality of all rational points in $R^3$

Question: Find the cardinality of the set of all points in $R^3$ all of whose coordinates are rational, and justify the answer. Idea: Call the set of all points in $R^3$ all of whose coordinates are ...
2
votes
1answer
42 views

cardinality of the set of points with one irrational coordinate, and one rational.

Find the cardinality of the set of all points in the plane which have one rational and one irrational coordinate. Justify you answer. My thoughts so far. We know that $\mathbb Q$, the set of all ...
1
vote
1answer
30 views

Cardinality of the set of minimal sets of a collection with cardinality $\aleph_0$

Let $X$ be a set and $F$ be a collection of subsets of $X$ such that $\vert F \vert = \aleph_0$ and let $F^*$ be the smallest collection of subsets of $X$ closed under intersection and complement (w.r....
11
votes
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 $\...
0
votes
1answer
19 views

Cardinality Proof Problem

Suppose A and B are sets such that card A ≤ card B. Prove that there exists a set C ⊆ B such that card C = card A. I know that there is only an injection from A to B. I'm having trouble showing that ...
0
votes
2answers
36 views

size of infinite strings and infinite alphabets

Please forgive the lack of formal vocabulary. Which set has a larger cardinality? A) a set of all possible countably infinite strings with a finite alphabet of symbols. B) a set of all possible ...
0
votes
1answer
20 views

Prove that $\{X \in P(Z)| X \text{ is finite}\}$ is enumerable. [duplicate]

I am not sure how to approach this problem. if you could help it would be great.
1
vote
1answer
32 views

How to proof or deproof that 2 amounts have the same cardinality?

I am new to cardinality proofs and so far I can't even understand exactly what I have to do here. Can anyone enlighten me? The task is the following: Let $M = \{ n \in \mathbb N^+ \mid n \mod 3 = 0 \...
6
votes
2answers
163 views

Prob 8, Sec 7 in Munkres' TOPOLOGY 2nd ed: How do we show these sets have the same cardinality?

Here's Prob. 8. Sec. 7 in Topology by James R. Munkres, 2nd edition: Let $X$ denote the two element set $\{0,1\}$; let $\mathscr{B}$ be the set of countable subsets of $X^{\omega}$. Show that $X^{...
0
votes
1answer
42 views

Prove that $(0,1) =c\mathbb R$

How do I prove this knowing that $f(x) = \tan(x\pi/2)$ is a bijection between $(0,1)$ and $(0, \infty)$? We also have a bijection between $(-1,1)$ and $(0,1)$.
3
votes
3answers
49 views

Finding the cardinalty of a subset of $\mathcal{P}(\mathbb{N}) $

I'm trying to find the cardinality of a certain set and I'm stuck. The problem is, we haven't learned about cardinality nor about any of its rules and equalities. We are asked to find a set whose "...
1
vote
1answer
71 views

Prob 6, Sec 7 in Munkres' TOPOLOGY, 2nd ed: The existence of an injection of a superset into the set means the sets have the same cardinality?

Let $A$ and $B$ be two sets such that $B \subset A$ and there is an injection $f \colon A \to B$. Then how to show that $A$ and $B$ have the same cardinality? Munkres' Hint: We define $A_1 \colon= ...
0
votes
4answers
79 views

How to create a bijection between $(0,1)$ and $(0, \infty)$?

I don't understand how to do this. The tip I have for the question is to first find a bijection between $(0,1)$ and $(1,\infty)$.
1
vote
1answer
243 views

Which of the following statements are true on countable sets

Show that the numbers of the form $\sum_{k=1}^{\infty} \frac{a_j}{3^k}$ , where $a_j = 0$ or $a_j = 1$ is countable . If $A = \cap_i^n A_1$ is countably infinite, then atleat one $A_i$ is counntable. ...
-1
votes
1answer
50 views

The cardinal number [closed]

Let $c$ be the cardinal number of $[0,1]$, i.e. $|[0,1]|=c$. Notice that $|A|\cdot|B| = |A\times B|$ and $|\mathbb{R}| = c$. Prove that $c\cdot c=c$. Don't use $ab=\max\{a,b\}$ where $a,b$ are ...
1
vote
1answer
84 views

Show that if X is an uncountable set and Y is a countable set then X $\bigcup$ Y has the same cardinality as X [closed]

Show that if X is an uncountable set and Y is a countable set then X $\bigcup$ Y has the same cardinality as X. Is this okay? $$$$ (X $\bigcup$ Y) = X + Y - (X $\bigcap $Y) $$$$ (X $\bigcup$ Y) + (X ...
1
vote
1answer
25 views

Equinumerousity of two sets

Given two infinite sets $A$ and $B$, I'm asked to show that the two sets $\mathcal{P}(B)^A$ and $\mathcal{P}(A)^B$ are equipotent. I proved it by showing that those two sets have the same cardinality (...
1
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0answers
20 views

Multisets and cardinality

Consider $0<\lambda<1$, and let $A$ be a multiset of positive integers. Let $A_n=\{a\in A: a\leq n\}$. Assume that for every $n\in\mathbb{N}$, the set $A_n$ contains at most $n\lambda$ numbers. ...
1
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0answers
14 views

Show cardinality problem with natural evens, all naturals and all integers

I know how to show the cardinality between 2 of the 3 could work. For example, you can establish a bijection with integers and all naturals through $\mathbb{N}$: s(n) = $\sum_{k =1}^n1$ and $\mathbb{Z}...
-2
votes
1answer
38 views

Prove that (0,1) ⊆ R and (4,10) ⊆ R have the same cardinality [duplicate]

Hows does (0,1) and (4,10) both existing as real numbers make it have the same cardinality?
0
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0answers
29 views

Power Set, Theory of Linear Algebra, and Axiomatic Systems

So for homework, my History of Math professor gave us these three questions: Explain why the power set of a set S (collection of all subsets) has the same cardinality as the set [all functions from ...
1
vote
1answer
31 views

Do the adeles $\mathbb{A}$ have the same cardinality as the real numbers $\mathbb{R}$?

My question today is about the Adeles and whether they have the same cardinality as the real numbers. Certainly $\mathbb{R}$ has more elements than $\mathbb{Z}$. This is by Cantor diagonalization. ...
1
vote
1answer
53 views

definition of a $\kappa$-small simplicial set

Let $\kappa$ be a regular cardinal. A set $X$ is $\kappa$-small, if $|X|<\kappa$. What does it mean for a simplicial set $X\colon \Delta^{op}\to Sets$ to be $\kappa$-small. I can imagine at ...
7
votes
2answers
550 views

Cardinality of the Euclidean topology and the axiom of choice

It is relatively straightforward to prove that the Euclidean topology has the same cardinality as the space itself. I have sketched a proof below. The proof seems to rely quite heavily on the axiom of ...
3
votes
1answer
61 views

Trying to understand the slick proof about the dual space

In this famous MO question, a beautiful proof is given of the fact $V\cong V^\ast\iff V$ is finite dimensional. I'm trying to go through it and I'm having some trouble. First of all, I know the in ...
1
vote
0answers
56 views

Examples of uncountable fields of characteristic $p$?

Let $p$ be a prime. The axioms of a field of characteristic $p$ is definable in first order logic and form a satisfiable theory $T$. Indeed, $T$ has arbitrarily large finite models and it also has an ...
1
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2answers
77 views

Are cardinal numbers well defined? Or could we have something like $\aleph_{1/2}$? [duplicate]

From what I understood, cardinal numbers are defined as: $\aleph_0$ = the cardinality of $\mathbb{N}$ $\aleph_{n+1}$ = is the least cardinal number greater than $\aleph_n$ The continuum hypothesis ...
0
votes
1answer
28 views

Cardinality of a metric space

Let $X$ be an infinite set .For any two metrics $d_1,d_2 $ on $X$ the identity map $i:(X,d_1)\to (X,d_2)$ is continuous. Prove that $X$ is always countable. I am not getting how to start this problem....
-1
votes
1answer
44 views

If a countable union of sets has card $\mathfrak{c}$, prove at least one of them has card $\mathfrak{c}$ [duplicate]

If $A=\bigcup_{n=1}^{\infty}A_n$ and $A$ has cardinality $\mathfrak{c}$, where $\mathfrak{c}$ is the cardinal of the continuum, prove that at least one of the $A_n$ has cardinality $\mathfrak{c}$.
1
vote
1answer
47 views

Comparing Hartogs number to set of all sets of relations on a set

Let $A$ be a set and let $\mathcal{H}(A)$ be the Hartogs number of $A$. Show that $|\mathcal{H}(A)|\neq|\mathcal{P}(\mathcal{P}(A\times A))$. Proof attempt: By contradiction. Suppose that $|\mathcal{...
0
votes
2answers
54 views

Show that $\leq$ is transitive over the cardinality of sets

$A, B, C$ are sets. I want to show $|A|\leq|B| \ \text{and} \ |B|\leq|C| \Longrightarrow |A|\leq|C|$. I am confused as to how I would approach this, because the sets in this problem can be either ...
6
votes
4answers
2k views

Taking away infinitely many elements infinitely many times [duplicate]

This is a somewhat hand wavy question but I'm not sure how to ask it more precisely. If we have a countably infinite sequence (or set), can we take away infinitely many elements from the sequence ...
1
vote
1answer
20 views

cardinal of the union of an increasing sequence

Let $A_n\subset \mathbb{R}$ for all $n$. If $\vert A_n\vert=C$ for all $n$ where $C$ is the cardinality of the real numbers and $A_n\subset A_{n+1}$. then Is it true that $\vert \bigcup_n A_n\vert=C$?...
0
votes
2answers
60 views

Show that the set is not countable

To show that a set is countable, you need to show 1 to 1 correspondence, right? So to test if it is 1 to 1 and also onto. So for this example: ...
1
vote
1answer
25 views

cardinality of the set of junctions intervals

Let P$=\{(a,b) : a,b \in \mathbb{R}\}$ and Q$=\{$ countable unions of elements of $P\}$. I 'm interested in knowing the cardinality of Q. I denote with $C$ if the cardinal real , then $C \leq \vert Q\...