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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Having trouble understanding aspects of cardinality.

I am having trouble understanding the meaning of $\omega_\alpha$, I thought it simply meant that it was the initial $\alpha$ segment of $\omega$, but then that wouldnt make sense if $\aleph_0$ is ...
2
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1answer
51 views

Attempt at proving the class of all cardinals is a proper class

Define $C=\{\alpha:\alpha=|x|$ for some set $x$$\}$ as the class of all cardinals. ($|x|$ being the cardinality of the set $x$) It will be enough to prove $C$ is a proper class by showing ...
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1answer
34 views

Cardinality of a line segment

If ${(a,b)}$ and ${(c,d)}$ are points in $ℝ^2$ then let $S$ be the set of point on the line segment that joins ${(a,b)}$ and ${(c,d)}$. Show $|S|= |ℝ|$ I can see this is similar to how the tangent ...
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4answers
78 views

Let $S$ and $T$ be sets such that $|S|=|T|$, prove that $|P(S)|=|P(T)|$

Let $S$ and $T$ be sets such that $|S|=|T|$, prove that $|P(S)|=|P(T)|$ where $P$ denotes a power set. From the theorem: If $S$ is a finite set with $n$ elements, then the cardinality of $P(S)=2^n$ ...
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0answers
41 views

What is the cardinality of the topology on $\mathbb{R}$? [duplicate]

Let $X = \mathbb{R}$ equipped with usual topology $\tau$, then $\tau = 2^\mathbb{R}$ Does this imply that the cardinality of $\tau$ is greater than continuum?
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0answers
70 views

Let a, b, c, d be real numbers such that a < b and c < d. Prove that |[a,b] x [c,d]| = c

Let $a, b, c, d$ be real numbers such that $a < b$ and $c < d$. Prove that $|[a,b]$ x $[c,d]| = c$. This $c$ on the end is the 'cardinality of the continuum' which means that the set has the ...
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0answers
21 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 ...
2
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1answer
24 views

What is the cardinality of the set of all $\sigma$-algebra containing $\mathcal{B}$

Let $X$ be a set, and $\mathcal{B}$ be a subset in $\mathcal{P}(X)$ the power set of $X$ Let $\{\mathcal{F}_i\}$ be the set of all $\sigma$-algebra containing $\mathcal{B}$ such that $\mathcal{B} ...
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37 views

Does log(aleph-null) have any meaning?

I'm familiar w/ the meanings and derivations of $\aleph_0$ and the general consequences of the continuum hypothesis (and the discussions at this question. ) So, if it turns out that $2^{\aleph_0} = ...
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0answers
27 views

Sets with cardinality strictly greater an c [duplicate]

Are there any examples of uncountably infinite sets with cardinality strictly greater than c other than the power set of the set of real numbers and the sequence of strictly larger sets obtained by ...
2
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0answers
38 views

Equal sets have power sets of equal order?

If two sets say $S$ and $T$ are equal is it true that $|2^{S}| =|2^{T}|$. Here is the motivation. Suppose that $S$ has infinite (or countable) order but that is is written as the union of a finite ...
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1answer
15 views

Cartesian and Inclusion-exclusion cardinality

Let $X,Y$ be finite sets such $\lvert X\rvert \leqslant \lvert Y\rvert,\\ \lvert X\cup Y\rvert = 16,\\ \rvert X\cap Y\rvert = 10,\\ \lvert X\times Y\rvert = 168$ Find: $\lvert X\times X\rvert$ ...
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1answer
45 views

linear independent set - $|Z|=|W|$

Let $V$ a nonzero vector space on a field $F$. Let $W$ and $Z$ two basis of $V$. If $|W|<\infty$, then $|Z|=|W|$. A hint is given by a certain textbook : Show that if $E$ is a basis on $V$ and ...
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0answers
36 views

For a set $A$, property P and cardinal $k$, when is the statement “$|A|<k$” equivalent to “Every function $A \to A$ satisfies P?” [closed]

Question in title. We know that a set $A$ being finite is equivalent to every injection from $A$ to itself being a bijection. Are there other cardinals $k$ such that every function from $A$ to itself ...
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1answer
17 views

Possible Cardinalities of the union of two sets

So the question is: What are the possible cardinalities of the union of the two sets $A$ (where $[A] = 5$) and $B$ (where $[B] = 9$) So, the smallest $[A \cup B]$ is when all elements of $A$ ...
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2answers
63 views

$\infty$ and $-\infty$ are to $\aleph_0$ / $\beth_0$ as “what” is to $\beth_1$?

So, I recently asked a question about whether $\beth_1$ had a negative, and I was promptly reprimanded because I confused $\aleph_0$ with $\infty$. Therefore, to help me understand the concepts ...
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1answer
52 views

Is noetherianity really about cardinals or ordinals?

If $\kappa$ is any cardinal, then one may define a "$\kappa$-Noetherian" ring as a ring such that for any module that has a generating set $S$ satisfying $|S|< \kappa$, then any submodule also has ...
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0answers
26 views

The weight of $X$ is $\aleph_0$ iff $X$ is second countable

Let $\omega(X)=\min\{|\mathcal{B}|:\mathcal{B} \mbox{ is a base of the topology of } X\}$. I want to make sure whether the following statement is true : $\omega(X)=\aleph_0 $ iff $X$ is second ...
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1answer
62 views

Set Theory: Tree Property

Why does the tree property hold for regular cardinals but not singular cardinals? (I.e. There exists a tree of height $\kappa$ with countable levels and no cofinal branch for $\kappa$ a singular ...
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49 views

Are the numbers $\beth_n$ for $n > 0$ signed or unsigned?

By extending the real number line in both directions, I know that $\infty$ or $\aleph_0$ or whatever else you want to call it has a negative, i.e. $-\aleph_0$ is a thing. Now, of course, $\beth_0 = ...
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1answer
48 views

What is the cardinality of the following set [closed]

Is the cardinality of the set {$x|$ $x ∈ N$ and $x=1.5$} infinite or not?
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47 views

How to measure difference in size of infinite objects?

We know $\Bbb R$ is bigger than $\Bbb Q$ because its cardinality is bigger. We know that $\Bbb R^2$ is bigger than $\Bbb R$, which is bigger than $[0, 1]$ because the latter can be thought of as a ...
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0answers
36 views

weight of a topological space

Let $\omega(X)=\min\{|\mathcal{B}|:\mathcal{B} \mbox{ is a base of the topology of } X\}$. In https://en.wikipedia.org/wiki/Base_(topology), it stated that if $\mathcal{B}$ is a basis of $X$, there is ...
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0answers
56 views

Cardinality of a group of permutation

Let $S$ be an infinite set of cardinality $\alpha$ and $G$ be a subgroup of $Sym(S)$. Let $\sigma(g)=\{s\in S \mid sg\neq s\}$ for each $g\in G$ and define $$Sym(S,\, \alpha)=\{g\in Sym(S)\mid ...
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1answer
27 views

Large ordered family of set inclusions complete?

I'm not an expert on set theory, so this might be trivial: Let $\dots \subseteq X_{\alpha} \subseteq X_{\alpha+1} \subseteq \dots$ be a chain of set inclusions, indexed by ordinals(!), s.t. ...
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1answer
52 views

Cardinality: Set of all binary sequence equal c

How do I prove the cardinality of the set of all binary sequences equal c? I know I have to find a bijective function from (0,1) to the set of all binary sequences. But I can't think of one. ...
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0answers
37 views

Cardinal of a set of numbers: naturals, integers, rationals and irrationals [duplicate]

My professor gave us these properties very fast in our class and I can't find a proper explanation for them, can someone help me please? (1) - The cardinal of the set of naturals is the same of the ...
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1answer
81 views

Does $|X|<|Y|$ imply $\mathcal{P}(X)<\mathcal{P}(Y)?$ [duplicate]

This might be a terribly simple question, but I cannot convince myself whether the answer is yes or no. Maybe I am missing something simple. I am not well-versed in the area of elementary set theory ...
3
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1answer
38 views

A Successor Cardinal is Regular

Trying to show that every cardinal $k$ , $k^+$ , its successor, is regular. This is what I've come up with. Thoughts? If this does not hold, then a cofinal map $f: \lambda\rightarrow k$ where ...
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1answer
50 views

Problem on infinite cardinal number

If $e$ is an infinite cardinal number and $d$ is a cardinal number satisfing $2 ≤ d ≤ 2^e$. I need to prove the following $$d^e= 2^e$$ Any help will be appreciated. Thank you in advance. .
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1answer
38 views

Cardinality of almost everywhere continuous functions

The cardinality of continuous real functions is $|\mathbb{R}|$ but I was wondering wether allowing functions to be almost everywhere continuous would increase the cardinality or not. On the one hand, ...
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0answers
38 views

Is it possible to define a group structure on arbitrary set? [duplicate]

Is it possible to define a group structure on arbitrary set? It is obvious for finite sets and also sets with cardinality |Q| and |R| and also we don't know that is there other cardinality betwen them ...
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1answer
42 views

for a given $\kappa$, find a model of DLO with a greater cardinality that has a dense subset of size $\kappa$

So given a cardinal $\kappa$, i am trying to find a dense linear order with no end points of cardinality greater then $\kappa$ which has a dense subset of size $\kappa$ I think I'm almost there What ...
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0answers
50 views

The supremum of any set of cardinals (considered as a set of ordinals) is again a cardinal.

An ordinal $\alpha$ is a cardinal iff no $\xi < \alpha$ is equivalent to $\alpha$. Now, let $A$ be any set of cardinals and $\sup(A)=\alpha$, then for $\xi < \alpha$ there is a $\beta \in A$ ...
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1answer
107 views

How does equality $\mathfrak{m}^\mathfrak{m} = 2^\mathfrak{m}$ in $\sf{ZF}$ relate to the axiom of choice?

Usually, the equality $\mathfrak{m}^\mathfrak{m} = 2^\mathfrak{m}$ for infinite cardinal $\mathfrak{m}$ is proved like that: $$ 2^\mathfrak{m} \leqslant \mathfrak{m}^\mathfrak{m} \leqslant ...
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4answers
111 views

Since $[0,1]$ and $\mathbb{R}$ are not homeomorphic, does that mean the cardinality of $[0,1]$ and $\mathbb{R}$ are different?

Given $[0,1]$ a closed interval on $\mathbb{R}$, we know that $[0,1]$ is compact and $\mathbb{R}$ is not, so these two spaces are not homeomorphic to each other. But homeomorphic perserves ...
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0answers
31 views

Cardinal Addition When At Least One is Infinite

Show that if at least one of κ > 0 and λ > 0 is infinite, then κ + λ = κλ = max{κ, λ}. My proof: Assume without loss of generality, κ > λ. If λ = 1, then by definition that at least one is infinite, ...
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1answer
25 views

The set of finite unions of intervals with rational endpoints is countable.

I don't know how to prove the following: Let $K:=\lbrace G : G$ is a union of finitely many intervals with rational endpoints$\rbrace$. Prove that $K$ is countably infinite. Here is my ...
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2answers
60 views

What is the result of a number greater than 2 raised to the power of {Aleph-0}?

So, I know that $2^{\aleph_0} = \beth_1$, right? What about another number, say $10$, raised to the power of $\aleph_0$? Is $10^{\aleph_0} = \beth_1$ also true, or is $10^{\aleph_0} > \beth_1$ ...
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1answer
23 views

Union of a chain of cardinalities?

I was trying to understand the union of a chain of cardinalities and I found this equation $$\kappa=\bigcup_{\alpha<\kappa} \alpha$$ for any cardinal $\kappa$ in the answers to this question. Can ...
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2answers
58 views

How many sequences of rational numbers in $[0,1]$ exist?

I was talking with a friend of mine and we wonder how many sequences of rational numbers on $[0,1]$ there exists. My first attempt was to consider that every sequence like that must be a subset of ...
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2answers
82 views

Proof of $\aleph_0^{\aleph_0} = \mathbb{c}$ without using Cantor's $2^{\aleph_0} = \mathbb{c}$

Prove that $\aleph_0^{\aleph_0} = \mathbb{c}$ without using Cantor's $2^{\aleph_0} = \mathbb{c}$ Card $\mathbb{N}^\mathbb{N} = \aleph_0^{\aleph_0}$ Card $(0, 1) = \mathbb{c}$ Define: $f: ...
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0answers
14 views

Family of infinite sets with finite intersections [duplicate]

I read somewhere that there exists a family of infinite sets $F \subset P_{inf}(\mathbb N)$, such that any two $X, Y \in F$ have a finite intersection and $\lvert F \rvert = \mathfrak c$. ...
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1answer
37 views

Where's the mistake in my reasoning?

Task: Find the cardinality of all such functions $f: P(\mathbb N) \rightarrow P(\mathbb N)$ that $f(\bigcup S) = \bigcup \lbrace f(Z) \mid Z \in S \rbrace$ The answer is: $\mathfrak c$ My ...
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2answers
63 views

Can you go from $\aleph_0$ to $\aleph_1$ with tetration or other higher order operators?

The paradox of Hilbert's Hotel shows us that you can not get past the cardinality of the natural numbers ($\aleph_0$) by adding a finite number (one new guest), adding an infinite quantity (infinitely ...
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1answer
46 views

Cardinal equality: $\;\left|\{0,1\}^{\Bbb N}\right|=\left|\{0,1,2,3\}^{\Bbb N}\right|$

I need to prove the above equality without Cantor-Bernstein Theorem or cardinals arithmetic (i.e., a bijection must be found). I know that for example $\;S\to 1_S=\;$ the indicator function, gives a ...
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1answer
189 views

Cardinality of the set of all bijections

Let $A$ be an infinite set and let $S$ be the set of all bijections $A \rightarrow A$. Then if $\mid A \mid = \kappa$, then $\mid S \mid = 2^\kappa$. I'm able to prove it for $A = \mathbb{N}$ by ...
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2answers
100 views

Why was $\aleph$ (aleph) chosen for infinities?

Why did Cantor choose a letter from the Hebrew alphabet to represent infinities, rather than using some Greek letter?
6
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1answer
241 views

Well-foundedness of cardinals and the axiom of choice

Without axiom of choice it is not generally true that the class of all cardinal (in this question we consider Scott cardinal rather than cardinals as ordinals) is not well-founded under the ordinary ...
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1answer
44 views

What result is $\left | \bigcup_{i \in I}A_i \right | =\sum_{i \in I} |A_i|$?

I'm reading a text that uses the following equality for disjoint sets $(A_i)_{i \in I}$: $$\left | \bigcup_{i \in I}A_i \right |=\sum_{i \in I} |A_i|$$ This has to do with disjoint unions, but I'd ...