This tag is for questions about cardinals and related topics such as cardinal arithmetics, clubs, stationary sets, cofinality, and principles such as $\lozenge$. Do not confuse with [large-cardinals] which is a technical concept about strong axioms of infinity.

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3
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0answers
65 views

Which Field Would You Use to Represent a Group Larger than $\aleph _1$?

I understand that in representation theory we try to understand a group $G$ by studying the homomorphisms $\rho\ \colon G \to $ GL$(V)$ where $V$ is a vector space over some field. I believe complex ...
1
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1answer
53 views

The set of all countably-infinite subsets of an infinite set

Let $A$ be an infinite set and $D(A)$ denote the set of all countably-infinite subsets of $A$ and let $P(A)$ denote ,as usual, the power set of $A$, then (i) does there exist a surjection of $D(A)$ ...
0
votes
1answer
56 views

Is the set of all sums-of-rationals-that-give-one countable?

Some (but not all) sums of rational numbers gives us 1 as a result. For instance: $$\frac12 + \frac12 = 1$$ $$\frac13 + \frac23 = 1$$ $$\frac37 + \frac{3}{14} + \frac{5}{14} = 1$$ Is the set of all ...
7
votes
2answers
187 views

Does $\mathcal P ( \mathbb R ) \otimes \mathcal P ( \mathbb R ) = \mathcal P ( \mathbb R \times \mathbb R )$?

Obviously the answer's yes if I replace $\mathbb R$ by a countable set. If $\mathbb R$ is replaced by a set $X$ having a greater cardinality, then the diagonal $\{ (x,x) , \ x\in X\}$ is not in the ...
4
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3answers
231 views

Infinite sets with cardinality less than the natural numbers

Are there any infinite sets that have a lower cardinality than the natural numbers? Is there a proof of this?
0
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0answers
49 views

Is this sentence OK?

I'm starting to write a paper. This is the sentence which I want to put first in the paper. It is well known that diagonal properties are useful in estimating certain cardinal invariants of a ...
-1
votes
1answer
36 views

Using only the def., how to show that for every cardinal there is a bigger one

How Can I deduce from Cantor's Theorem that for every cardinal $\alpha$ there is a cardinal $\beta> \alpha$. A cardinal is an ordinal which is equal to its cardinality.
2
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1answer
47 views

Infinite cardinal comparation

Let $\alpha$ and $\beta$ two infinite cardinal numbers. Can we have $\alpha = \beta^\alpha$? This problem comes from a situation where I am dealing with the cardinal of a set of functions.
2
votes
1answer
56 views

Number of models for some theory

Let $\mathcal L = \{ E(\_,\_) \}$ and $T$ be the $\mathcal L$-theory that says that $E$ is an equivalence relation with an infinite number of infinite classes. (I find this statement not clear, ...
2
votes
3answers
46 views

about definition of a cardinal

Definition: the cardinality of a set $A, |A|$ is the least ordinal s.t. $A \sim \alpha$ Definition: We define a cardinal to be an ordinal $\alpha$ s.t. $\alpha = |\alpha|.$ i.e, an ordinal s.th. ...
0
votes
1answer
56 views

Cardinals and $2^\aleph$

The cardinal of $\{e^{ax}\,\, |\,\, a \in \mathbb{R}\}$ is $\aleph$, What is the cardinal of the group of all functions that are linear combinations of ones from the first group? If it's not ...
3
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1answer
53 views

Banach space with cardinality bigger than $\mathfrak{c}$.

By using the infromation contained in this post, we have that the cardinality of every Banach space is equal to its dimension, which in turn, is bigger or equal to $\mathfrak{c}$. In my area of ...
7
votes
1answer
148 views

weak consequence of GCH

Can ZFC prove that there is a regular uncountable cardinal $\kappa$ such that $2^{<\kappa} < 2^\kappa$? Note, if the answer is no, it would require a strong global violation of SCH, so large ...
10
votes
3answers
292 views

Can sets of cardinality $\aleph_1$ have nonzero measure?

$\aleph_1$ is the cardinality of the countable ordinals. It is the least cardinal number greater than $\aleph_0$, and assuming the continuum hypothesis it's equal to $\mathfrak{c}$, the cardinality of ...
2
votes
1answer
69 views

How do I show that $\kappa^+ \le 2^\kappa$ for every cardinal $\kappa$?

I'm looking into using $2^\kappa = \kappa^\kappa$ to prove this because it follows from $2 < \kappa < \kappa^+$ but I'm not sure how to connect the two together. Anyway, this might not even be ...
1
vote
1answer
63 views

Cartesian product of large sets

For a non-empty set $A$ let $A'$ denote the Cartesian product of $A$ with itself taken denumerably many times. Now given a set $S$ whose cardinality is strictly greater than the cardinality of ...
1
vote
1answer
70 views

Maximum cardinality

Let $X$ be some set and $P$ be some subset of ${\frak P}(X)$. We can define the smallest cardinal $\frak k$ such that any $A \in P$ has cardinal $\leq \frak k$. Indeed we can consider the set of all ...
0
votes
1answer
47 views

Finding a limit using arithmetic over cardinals

What is the value of: $$\lim_{n \to \infty} \frac{n}{2^n} (n \in \mathbb{N})$$ It seems to me that I can use L'Hopital's rule, but does that rule take into account types of infinity? More precisely, ...
0
votes
1answer
36 views

$L$ is a class of languages that cannot be represented by a regular expression. How to state cardinality of $L$.

$L$ is a class of languages that cannot be represented by a regular expression. The book says that the cardinality of $L$ is $2^{\aleph_0} > \aleph_0$ what's the logic behind getting the ...
0
votes
2answers
64 views

Calculate the cardinalities of $A\times D$, where $D$ is an infinite set?

please help me with this one? $A = \{0,1,\{2,3,4\}\}$ $D = C \times N$ where $C = \{1,5\}$ and $N$ is natural numbers need to calculate $A \times D$? any help would be greatly appreciated
1
vote
1answer
55 views

Hartogs' proposition

I want to prove that if "there are not well-ordered sets then there are incomparable sets. In fact if $a$ is not a well-ordened set then there is an ordinal $\alpha $ such that neither $\alpha ...
1
vote
1answer
43 views

Does there exist an injection from $P(S)$ to $u(S)$

Let $S$ be an uncountable set , let $u(S)$ denote the set of all uncountable subsets of $S$ and let $P(S)$ denote , as usual , the powerset i.e. the set of all subsets of $S$, then does there exist an ...
2
votes
3answers
100 views

Discarding lower cardinality subset doesn't change infinite cardinality?

This is a basic question about set theory. I am of the belief that if $A$ is a set of some infinite cardinality and $B$ is a subset with lower cardinality, $A\setminus B$ has the same cardinality as ...
2
votes
1answer
127 views

Prove Cardinality of Power set of $\mathbb{N}$, i.e $P(\mathbb{N})$ and set of infinite sequences of $0$s & $1$s is equal.

I tried doing this by defining a function $f$ which takes an element (a subset of $\mathbb{N}$) and maps it to an infinite sequence of $0$s & $1$s. A subset, i.e the element of $P(\mathbb{N})$ ...
29
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7answers
1k views

Refuting the Anti-Cantor Cranks

I occasionally have the opportunity to argue with anti-Cantor cranks, people who for some reason or the other attack the validity of Cantor's diagonalization proof of the uncountability of the real ...
5
votes
1answer
51 views

What is the cardinality of an element of an free ultrafilter?

Let $U$ be a free ultrafilter on a set $X$. I want to prove that the cardinality of every element $u\in U$ is equipotent to $X$. Is that true? Or does it lack some hypothesis?
0
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2answers
68 views

Proove formally that |N| = | N union a finite set |.

I'd like to show that the cardinality of $\mathbb{N}$ is the same as the cardinality of $\mathbb{N}$ union some other finite set (disjoint from $\mathbb{N}$) For e.g show that : $|\mathbb{N}|= | ...
2
votes
1answer
78 views

Show that 2S = S for all infinite sets

I am a little ashamed to ask such a simple question here, but how can I prove that for any infinite set, 2S (two copies of the same set) has the same cardinality as S? I can do this for the naturals ...
0
votes
1answer
57 views

prove $| \cup_n A_n| <\mathfrak{c}$

If $\{A_n: n \in \mathbb{N} \}$ is a sequence of subsets of $\mathbb{R}$ and $|A_n| < \mathfrak{c}$ for all $n$. Prove $| \cup_n A_n| <\mathfrak{c}$ with $\mathfrak{c}$ the cardinality of ...
0
votes
1answer
61 views

Are there countably many infinities?

$\aleph_0$, $\aleph_1, \aleph_2$ and so on are indexed by a natural number so shouldn't there be countably many infinities?
2
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5answers
337 views

Proof that aleph null is the smallest transfinite number?

The wikipedia page on the cardinal numbers says that $\aleph_0$, the cardinality of the set of natural numbers, is the smallest transfinite number. It doesn't provide a proof. Similarly, this page ...
2
votes
1answer
36 views

$\sum_{b\prec a}\sigma_b\prec\prod_{b\prec a}\delta_b$ [duplicate]

Suppose $a$ is a cardinal number and $\lbrace\sigma_b\rbrace_{b\prec a}$ and $\lbrace\delta_b\rbrace_{b\prec a}$ are two families of cardinal numbers such that for all $b\prec a$, ...
2
votes
1answer
59 views

cardinality of the set of all dense subsets of $\Bbb R$

Let $$A=\{X \subseteq \mathbb R : \operatorname{cl}(X)=\mathbb R\}$$ Prove that the set $A$ and $P(\mathbb R)$ have the same cardinality. Well, the first thing it came to my mind was the injective ...
2
votes
3answers
89 views

Is this set of integer sequences countable?

I'm faced with a set of strictly increasing functions $\Bbb N\to \Bbb N$, i.e. positive integer valued sequences. The only thing I know about them is that they are pairwise eventually disjoint, by ...
2
votes
1answer
32 views

If $\aleph_\lambda>2^{\aleph_0}$ is a limit cardinal, then $\aleph_\lambda^{\aleph_0}=\aleph_\lambda$?

I know that for successor cardinals, the result holds; i.e. if $\aleph_\alpha$ is a successor cardinal, then because this implies that it is a regular cardinal, then we can use this to show that if ...
3
votes
2answers
81 views

How can I quantify over the class of all cardinalities?

I'd like to quantify over all cardinalities of sets. My end goal is to make a category-theoretic arguement: For all cardinalities of sets, in the category of sets with maps as morphisms: the ...
2
votes
4answers
202 views

Prove that $(0,1)$ is cardinally equivalent to $[0,1)$

How's this done? Also, I am wondering, are all subsets of $\mathbb{R}$ cardinally equivalent to each other? If not, why not?
3
votes
1answer
86 views

A question dealing with finite unions and type.

I've been independently reading Kunen's newest set theory book for a self-study course. I'm looking at his chapter on cardinal arithmetic, thoroughly reading proofs and working on exercises. After an ...
4
votes
1answer
65 views

Proving that $\sf Add$$(\aleph_\omega , 1)$ collapses cardinals $\leq \aleph_\omega$

First, let me fix some notation. $\sf Fn$$(I, J, \kappa) = $ the poset of all partial functions $p$ such that $|p| < \kappa$, dom$(p) \subseteq I$ and rng$(p) \subseteq J$. $\sf Add$$(\kappa, ...
6
votes
2answers
127 views

In ZF, how would the structure of the cardinal numbers change by adopting this definition of cardinality?

In ZFC, a good way of ordering sets by cardinality is by leveraging the notion of an injection. We define: $$X \lesssim Y \leftrightarrow \mbox{ there exists an injection } X \rightarrow Y.$$ ...
3
votes
2answers
143 views

Two exercises on set theory about Cantor's equation and the von Neumann hierarchy

Good evening to all. I have two exercises I tried to resolve without a rigorous success: Is it true or false that if $\kappa$ is a non-numerable cardinal number then $\omega^\kappa = \kappa$, where ...
2
votes
2answers
100 views

How do the terms “countable” and “uncountable” not assume the continuum hypothesis?

Every countable set has cardinality $\aleph_0$. The next larger cardinality is $\aleph_1$. Every uncountable set has cardinality $\geq 2^{\aleph_0}$ Now, an infinite set can only be countable or ...
3
votes
2answers
600 views

Proof for the relationship between Cardinality of Natural numbers and Cardinality of Real Numbers

(I am a 13 year old so when you answer please don't use things that are TOO hard even though I actually can understand quite complex stuff) I was studying Infinite sets and their cardinality (not in ...
5
votes
3answers
495 views

Prove that transcendental numbers exist: Are there less paniful ways of doing it?

I've found this exercise on Boolos' Logic and Computability: A real number $x$ is called algebraic if it is a solution to some equation of the form: $$c_{\small d}x^{\small ...
2
votes
1answer
104 views

If $(X,<)$ is an uncountable well-ordered set whose sections are countable, $f:X\to X,f(x)<x$, then $\exists x,f^{-1}(x)$ uncountable.

For an uncountable set $X$, with a well-ordering $<$ (i.e. every subset of $X$ has a minimum element), and a property that for any $x\in X$, $\{y\in X|y<x\}$ is countable, we have a function ...
0
votes
1answer
34 views

Cardinality of given set

Given, $$A=\{B\subset \mathbb{N}: B \text{ is finite} \vee B^c \text{ is finite }\}$$ How can I prove that A is countable. For me it seems it is uncountable.
1
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1answer
92 views

Calculating cardinal numbers of subsets in $\mathbb R\times\mathbb R$

Calculate the cardinal numbers of the following subsets of $\mathbb R\times\mathbb R$ : a.$X=\left\{ (a,b)\in\mathbb{R}\times\mathbb{R}\mid a+b\in\mathbb{Q}\right\} $ b.$Y=\left\{ ...
0
votes
1answer
21 views

Name for the sense of how many items are present

Sorry, this might be slightly off topic: there's a word for the ability to look at a small set of items are know how many are there without counting them, but I can't remember what it is and I can't ...
9
votes
4answers
259 views

Infinite sets and their Cardinality

(I am a 13 year old so when you answer please don't use things that are TOO hard even though I actually can understand quite complex stuff) I was studying Infinite sets and their cardinality (not in ...
1
vote
2answers
102 views

Cardinality and bijections

i am following a course in axomatic set theory. We are talking about bijections, injections, Schröder-Bernstein-theorem, etc. at this moment. I want to make te following exercises: Prove: ...