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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30 views

Validity of certain arguments about the countability of infinite sets

I am trying to get an understanding, in layman's terms / on an intuitive level, why some arguments about the countability of infinite sets are valid, and some arguments which seem almost identical on ...
7
votes
1answer
186 views

Showing a cardinal is regular

I've been thinking about this question, but to no avail and I've got to ask. How to show that for $\kappa\geq\aleph_0,$ $\mu=\min\{\lambda: \kappa^{\lambda} > \kappa\}$ is regular? If I wanted a ...
3
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2answers
36 views

Another characterization of the cofinality?

Is it true that $cf(\kappa)=\min \{\lambda:\ \kappa^{\lambda}>\kappa\}$? $cf(\kappa)$ is certainly $\geq$ than that minimum since $\kappa^{cf(\kappa)}>\kappa$, but I don't know how to tackle ...
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2answers
100 views

What are some applications of large cardinals?

Most mathematicians don't often encounter cardinalities larger than that of the continuum, it seems? What are some results outside of pure set theory/logic that rely on the properties of larger ...
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0answers
35 views

Equivalence Relations and Cardinality

I'm looking at the question below from a past paper: What is an equivalence relation? Say that two sets $X$ and $Y$ are related via the relation $\rho$ if $X$ and $Y$ have the same cardinality. Prove ...
2
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1answer
71 views

Without the Axiom of Choice, $\aleph_0<2^{\aleph_0}$ implies $\aleph_1\le 2^{\aleph_0}$?

Question: In ZF (so AC does not necessarily hold) does the following claim hold? $\aleph_0<2^{\aleph_0}$ implies $\aleph_1\le 2^{\aleph_0}$ This question arose to me when reading the top ...
3
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1answer
19 views

Comparability of a set and a subset of power set.

It's well known that for any set $A$, $A < P(A)$. But now, I have some question that, WITHOUT AC, can we guarantee that $A \leq X$ or $X \leq A$ whenever $X \subseteq P(A)$? Thank you.
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1answer
51 views

Equinumerousity of operations on cardinal numbers

I want to prove for all Cardinal numbers $a$, $b$, $c$ that: $(a \cdot b)^c =_c a^c \cdot b^c$ $a^{(b+c)} =_c a^b \cdot a^c$ $(a^b)^c =_c a^{b \cdot c}$ I know that for 1. it's enough to show ...
3
votes
3answers
63 views

Question about $\aleph$-fixed point

I am working through a proof on cardinals I found and can't reason some of the steps. The proposition is that there is an $\aleph$-fixed point, i.e. there is an ordinal $\alpha$ (which is ...
2
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1answer
104 views

Regarding the axiom $2^\kappa = 2^{\kappa^+}$ for regular cardinals $\kappa$, and its relationship to a couple of other axioms.

(Take ZFC as background.) The following two statements both follow from GCH: ICF. Injective continuum function. The continuum function (i.e. $\kappa \mapsto 2^\kappa)$ is injective. NJA....
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2answers
49 views

What is $n^{\aleph_0},n\in\mathbb N$

Can I say that $$n^{\aleph_0}=2^{\aleph_0\log_2n}=2^{\aleph_0}=\aleph_1$$
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1answer
68 views

Does $\operatorname{card}(X) < \operatorname{card}(Y)$ imply $\operatorname{card}(X^2) < \operatorname{card}(Y^2)$ without choice?

I looked to see if this question was already posted, but did not find anything. Please let me know if this is a duplicate. Assume $X, Y$ are infinite sets such that there is an injection $X \to Y$ ...
0
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1answer
37 views

Number of subsets with the same cardinality

Suppose we have a set $S$ with cardinal number $n$, such that $n+n = n$. Consider the set, T, of all subsets with cardinality $n$. How can I show that the cardinality of $T$ is $2^n$? (Without the ...
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1answer
19 views

Constructing $2^\kappa$ vectors with a certain property

Take an infinite rectangular array with $\kappa$ columns and $2^\kappa$ rows where $\kappa$ is some infinite cardinal. Can you fill it up with at most $\kappa$ different elements in such a way that ...
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2answers
42 views

Proving cardinality of an uncountable sum

I'm trying to prove the following thing: For a family $\mathcal{A}$ of countable sets such that $|\bigcup\mathcal{A}|$ is uncountable and such that $\big|\{A\in\mathcal{A}: x\notin A\}\big| \leq\...
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1answer
36 views

Set of increasing functions from N to N is uncountable

I know it is uncountable. But what is wrong with this proof? I use the lemma that a countable union of countable sets is countable. Let $f(0)=0$. Then for each function $f$, construct $[a_0, a_1, ...
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0answers
37 views

Perfect Sets of real numbers have the same cardinality as the reals

I am currently trying to understand a proof from here that all perfect sets have the same cardinality as $\mathbb{R}$. So given some perfect set $P \subseteq \mathbb{R}$, the identity mapping $\text{...
0
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1answer
23 views

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 $\...
1
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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
votes
1answer
71 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
44 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 ...
0
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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 ...
0
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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
27 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
23 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
39 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
38 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 ...
0
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4answers
89 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
42 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
99 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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1answer
31 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
41 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
45 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 ...
1
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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=...
5
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1answer
22 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 $A$ ...
1
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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 ...
5
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1answer
56 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 ...
3
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1answer
74 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
54 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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0answers
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
38 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 ...