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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Cardinality of the union and product of two sets without AC

We have the following results Let $A$ and $B$ be infinite sets s.t. $|A|=|B|$, then $|A\cup B|=|A|$. I was wondering if we can prove that without the Axiom of Choice or without using cardinal ...
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2answers
45 views

Cardinal of an infinite set

In our course about combinatorics, our maths teacher recently introduced to us the notion of cardinality with the following definition: Let $E$ be a set. If there exists an integer $n$ and a ...
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Minimal Possible Ordinal for a Cardinal

For a given $\aleph_\alpha$, what is the minimal ordinal possible such that $|\beta| = \aleph_\alpha$? More precisely, assume we have a model $N$ of ZFC. $N$ could be an extension of many different ...
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2answers
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More numbers between $2$ and $4$ than between $2$ and $3$? (I am not a mathematician.) [duplicate]

Between $2$ and $3$ there are infinite numbers and between $2$ and $4$ there are infinite numbers. So which "infinity" is greater?
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1answer
46 views

Need a formal proof?

If A and B are two equipotent sets (they have 1-1 correspondence). Prove that if A is denumerable then B is also denumerable. It is easy to understand by intuition. But I can't understand how to ...
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1answer
62 views

What do you call a cardinal $\kappa$ that is a limit of $\kappa$-many cardinals?

For instance, $\omega$ is the limit of $\omega$-many cardinals. But of course $\omega_1$ is not the limit of $\omega_1$-many cardinals. 1) Are there cardinals other than $\omega$ with this property? ...
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3answers
174 views

Ordinal with given cardinality (without AC)

Is it possible to show that every cardinality has an ordinal with this cardinality (without the axiom of choice)? If so, how?
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2answers
192 views

Can every set be expressed as the union of a chain of sets of lesser cardinality?

If a set $S$ has countably many elements $\{x_n\}$, it can be expressed as a union of a chain of finite sets $$ \{x_1\} \subset \{x_1,x_2\} \subset $$ But what about a set of arbitrary cardinality ...
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1answer
56 views

Is this function one to one? Why? [closed]

Is the function $f:P(\mathbb{N})\to\mathbb{R}$ defined by $$f(A)=\sum_{n\in A}\dfrac{2}{3^{n+1}},\quad\forall\,\,A\in P (\mathbb{N}),$$ an one to one function? Please help understand why. For me, ...
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2answers
41 views

Cardinal of a difference of power sets

How can the following be calculated? Given the sets $X = \{1, 2, \dots, 10\}$ and $Y = \{1, 2, \dots, 12\}$, compute $| \mathcal P (Y) \setminus \mathcal P (X) |$, where $\mathcal P (X) = \{ A \mid A ...
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2answers
284 views

Where is the flaw in my Continuum Hypothesis Proof?

I am not a mathematician, but rather a computer engineer with a curious mind. The continuum hypothesis (CH) has gripped my attention today, and I even asked a question about it earlier today. ...
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1answer
85 views

Is this interpretation of the continuum hypothesis correct?

I am not a mathematician, but rather a computer engineer with a curious mind. The continuum hypothesis states (I believe) that there does not exist a set $S$ such that $\aleph_0 < |S| < ...
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2answers
75 views

Why are Aleph numbers by definition of the form $2^x$?

The first Aleph number is $\aleph_0$, and my question is this: why is the second Aleph number defined to be $\aleph_1 = 2^{\aleph_0}$? If I remember correctly, it had something to do with power sets ...
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2answers
57 views

Cardinality of the set of all real functions which have a countable set of discontinuities

I'm having a trouble calculating the cardinality of the set of all functions $f:\mathbb{R} \longrightarrow \mathbb{R}$ which have at most $\aleph_0$ discontinuities (let's call the set $M$). A hint ...
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1answer
51 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 ...
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2answers
56 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 ...
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3answers
937 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 ...
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0answers
40 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 ...
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3answers
94 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? ...
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1answer
178 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 ...
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1answer
201 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 ...
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1answer
32 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 ...
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2answers
40 views

Applying power set finite times

Is every infinite set $A$ smaller than a set of the form $\mathcal P (\mathcal P(\dots \mathcal P(\mathbb N)))$?
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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 ...
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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 ...
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2answers
77 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 $\mathbb{R}$. ($|N| < |C| \leq \mathbb{R}$) ...
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4answers
41 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 ...
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1answer
20 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 ...
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0answers
28 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 ...
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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 ...
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1answer
39 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 ...
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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 ...
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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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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 ...
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2answers
33 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 ...
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1answer
19 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.
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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 ...
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2answers
160 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 ...
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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 $(0,1)$.
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3answers
47 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 ...
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1answer
67 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= ...
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4answers
75 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 $(1,\infty)$.
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1answer
242 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. ...
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1answer
46 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 ...
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1answer
73 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 ...
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1answer
24 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 ...
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0answers
17 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. ...
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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 ...
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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?
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28 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 ...