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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The fundamental group of a wedge sum of countably many circles is countably generated, hence countable.

The fundamental group of a wedge sum of countably many circles is countably generated, hence countable. I don't really understand this statement. How can I see that the free product of countable $\...
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1answer
40 views

$y\mapsto 10^y$ Bijection between integers and powers of 10

Say $y$ and $x \in \mathbb N$ such that $10^y = x$ MSE research gives me this: $y\mapsto 10^y$ For some of you to see the bijection between the integers and powers of 10 will be obvious, but I ...
5
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1answer
70 views

Continuum hypothesis : Number of connected components of $A^c$ in $\Bbb R$

Let $A \subset \Bbb R$ with a countably infinite number of connected component (for the usual topology). What can be the cardinal of the set of the connected components of $A^c$? With continuum ...
2
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1answer
41 views

Exponential of cardinal numbers

there is two wrong statement that I want to find counterexample for them. if $\alpha$ and $\beta$ and $\gamma$ be infinite cardinals then show that these two statements are wrong $\alpha < \beta \...
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1answer
141 views

Show that the set of powers of ten is countably infinite

My question asks: An $x \in \mathbb{N}$ is called a power of ten if there exists a $y \in \mathbb{N}$ such that $10^y = x$. Show that the set of powers of ten is countably infinite. I understand that ...
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1answer
23 views

How to create a bijection from the union of the set natural numbers and square root 2 to the set of natural numbers?

Since $\Bbb N$ and ${\sqrt2}$ are each countable sets, I see that the union is also countable. From this and the fact that $\Bbb N$ union ${\sqrt2}$ is infinite we know there exists a bijection to $\...
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2answers
12 views

Finding the cardinality of a collection of lines?

I'm trying to find the cardinality of the set of lines such that the lines are not vertical, the slopes are a natural number, and the line has a natural number for a $y$-intercept. Because there are ...
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1answer
14 views

Showing the cardinality of a bounded shape in the xy plane?

I am trying to show that the cardinality of the space between $x^2+y^2<1$ and $x+y>1$ is the same as the cardinality of real numbers I haven't a clue where do begin with something like this. I ...
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1answer
25 views

Cardinal exponentiation question of infinite cardinals.

I got a little confused with a question about cardinal exponentiation: Let $\beta$ be an ordinal, and let $K_\alpha$ , $\alpha < \beta$, be infinite cardinals with $K= \sum_{\alpha<\beta}K_\...
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0answers
22 views

Impossibility of constructing a continuum-size linearly independent set in $\Bbb R$ [duplicate]

This is a response to the following exchange at Is there a quick proof as to why the vector space of $\mathbb{R}$ over $\mathbb{Q}$ is infinite-dimensional? [Bill constructs a $\aleph_0$ ...
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1answer
38 views

Construct an injective function $f:[a,b]\times[c,d]\rightarrow\mathbb{R}$ for some $a,b,c,d\in\mathbb{R}$

Let $a,b,c,d\in\mathbb{R}$ such that $a<b$ and $c<d$ be given. Construct an injective function $f:[a,b]\times[c,d]\rightarrow\mathbb{R}$. My intuition is to construct a function $g:[a,b]\times[c,...
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0answers
32 views

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
55 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 $On\...
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1answer
36 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
87 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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93 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
26 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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0answers
40 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
44 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 ...
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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
19 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 $X=...
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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
20 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
67 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
55 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
27 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
73 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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0answers
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
52 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
60 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 |\sigma(...
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1answer
28 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. $X_{\alpha}...
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1answer
59 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. Cantor'...
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0answers
38 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 ...
3
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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
50 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
52 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
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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 ...
3
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0answers
54 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
110 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 (2^\...
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4answers
114 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
32 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
26 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 approach:...
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2answers
64 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 $10^{\aleph_0} > \beth_1$ ...