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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1answer
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Preserve Cardinals and Adding No Bounded Subsets

In Chapter 15 at the bottom of page 228 of $\textit{Set Theory}$ by Jech, he writes that if $\kappa$ is a cardinal in $V$ and if $\kappa$ has no new bounded subsets in $V[G]$, then $\kappa$ remains a ...
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4answers
306 views

Looking for a problem where one could use a cardinality argument to find a solution.

I would like to find an exercise of the type: Find some $x$ in $A\setminus B$. Solution: since $A$ is uncountable and $B$ is countable such $x$ exists...
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1answer
53 views

How many ways can we totally order a set $X$ (but not necessarily the whole set $X$, perhaps just a subset) up to isomorphism?

Here's Attempt #2 at asking this question. Suppose $X$ is a set (not necessarily finite), and let $T$ denote the collection of all totally ordered sets $(A,\leq)$ such that $A \subseteq X.$ Now let ...
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3answers
42 views

Prove that $\{(a,b):a,b\in\mathbb N, a\geq b\}$ is denumerable.

If $S=\{(a,b):a,b\in\mathbb N, a\geq b\}$, how do I prove that $S$ is denumerable? Work: Since $S \subseteq\mathbb{N\times N}$ I know that $S$ is denumerable. But I don't know how to structure the ...
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0answers
43 views

General question: what happens if we replace the regularity stipulation in GCH with other conditions?

I went to bed last night pondering the following. We can formulate both a weak and a strong generalized continuum hypothesis. GCH0. If $\kappa$ is an infinite cardinal number, then $2^\kappa$ is ...
2
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2answers
64 views

Proving that (0,1) and [0,1] are numerically equivalent.

as the title suggests, I need help proving that the cardinality of $(0,1)$ and $[0,1]$ are the same. Here is my work: $f:[0,1] \rightarrow (0,1)$ Let $n\in N$ Let $A=\{\frac{1}{2}, \frac{1}{3}, ...
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2answers
36 views

cardinality with finite sets

$A,B,C$ are finite sets. Suppose $A\subseteq B \subseteq C$ and $\#A=\#C$. Prove that $\#A=\#B$ and $\#B=\#C$. Should I prove this by showing that there exist an element in $A$ that exist in $B$ and ...
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2answers
38 views

Cardinality of $\lim_{k\to\infty}\mathbb N^k$ vs. $\mathbb N^\infty$

My friend and I are having a disagreement over whether the number of terms in the following series is countable or uncountable: $$\sum_{i=1}^\infty a_i + \sum_{i=1}^\infty\sum_{j=1}^\infty a_{ij}+ ...
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0answers
15 views

If $X$ is a finite set of cardinality $n$, where $n$ exists in $P$, show that the following conditions on a function $f: X \to X$ are equivalent: [duplicate]

(a) $f$ is an injection (b) $f$ is a surjection (c) $f$ is a bijection I know that (c) implies (a) and (b) and (a) and (b) imply (c). I also have the following definition that I've been playing ...
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1answer
59 views

$\ell^p(I)$ space and a dense set of this space

Let $I$ be an infinite set and $1\leq p<\infty$. Show that $\ell^p(I)$ has a dense set of the same cardinality as $I$. For this I put $$X=\{(x_i); x_i\in \Bbb C , x_i=0 \text{ for all but ...
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2answers
35 views

Cardinality of two sets cross-multiplied

Let $A$ and $B$ be sets. Prove that $ \#(A \times B) = \#(B \times A)$. What I have done: There exist an element $m$ in $A$ such that the element also exists in $B$. If $\#A = \#B$, then $\#B = ...
4
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2answers
42 views

Proving that $\mathrm{card}(2^{\mathbb{N}})=\mathrm{card}(\mathbb{N}^\mathbb{N})$

I'd like to prove that $\mathrm{card}(2^{\mathbb{N}})=\mathrm{card}(\mathbb{N}^\mathbb{N})$, I have the following 'sketch' but I'm not sure if this works. ...
14
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1answer
1k views

Overview of basic results on cardinal arithmetic

Are there some good overviews of basic formulas about addition, multiplication and exponentiation of cardinals (preferably available online)?
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4answers
673 views

What's the cardinality of all sequences with coefficients in an infinite set?

My motivation for asking this question is that a classmate of mine asked me some kind of question that made me think of this one. I can't recall his exact question because he is kind of messy (both ...
1
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1answer
72 views

What is the cardinality of $X$?

Let $X=\{x\in D^\mathfrak{c}: 0<|\{\xi<\mathfrak{c}:x(\xi)=1\}|\le\omega_1\}$, where $D=\{0,1\}$. What is the cardinality of $X$? I think it is $\mathfrak c$, however I'm not sure. Also I don't ...
0
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1answer
22 views

An alternative succinct proof needed for trivial cardinality fact

Let $|X|$ denote the cardinality of a set, i.e. the least ordinal $\alpha$ such that there is a bijection between X and $\alpha$. For any sets $X$ and $Y$ we write $X\preccurlyeq Y$ if the exists an ...
2
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1answer
85 views

What are interesting examples of existential proofs based on cardinality arguments?

Probably the most famous example of a proof, where consideration of cardinalities is used to show existence of some object, it the Cantor's proof that there exist transcendental numbers. What are ...
1
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1answer
67 views

Bigger infinity than real number infinity [duplicate]

Is there a bigger infinity than the infinity of cardinality of the real numbers $R$ ? i.e. is there a set to which real numbers can't be mapped one-one to ?
0
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3answers
52 views

Are the following sets countable?

I'm trying to determine if the following sets are countable: (a) $\mathbb{Z}^{[0,1]}, (b) [0,1]^{\mathbb{Z}}, (c) \mathbb{Z}^{\mathbb{Z}}$, (d) the set given by functions $f:\mathbb{Z}\to\mathbb{R}$ ...
0
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1answer
23 views

Cardinal arithmetic basics

Let's say we have $\omega + \omega$. Since these sets are not disjoint we can replace them by disjoint sets of the same cardinality, namely $\omega \times \{0\}$ and $\omega \times \{1\}$. Then $\big ...
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2answers
76 views

Does $a \le b$ imply $a+c\le b+c$ for cardinal numbers?

Let $a, b, c$ be cardinalities. Prove or disprove: If $a \le b$ then $a+c\le b+c$ I realize that $a \le b$ means that there's a bijection between A and B. But I don't really know what ...
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1answer
17 views

Cardinality of rational exponentiation orbit space

Let $X=(0,\infty)$ be the set of positive real numbers. Let $G=\mathbb{Q}\backslash\{0\}$ be the multiplicative group of rational numbers. $G$ acts freely on $X$ by exponentiation: $r\cdot x=x^r$ for ...
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0answers
43 views

What is the cardinal number of the following?

What is the cardinal number of $\{f:\mathbb{R}\longrightarrow \mathbb{N}\ |\text{ $ f$ is an injective function}\}$?
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0answers
79 views

Solovay Theorem

Solovay's Theorem states If $\kappa$ is regular uncountable, then any stationary set in $\kappa$ can be partitioned into $\kappa$-many pairwise disjoint stationary sets. For regular uncountable ...
5
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1answer
68 views

Existence of a real uncountable $\aleph_{\alpha}$ without $\mathsf{AC}$

Set theory (Jech) $\text{p.}\;27:$ It is an open problem whether one can prove without the axiom of choice that there exists a regular uncountable $\aleph_{\alpha}\;($the informed guess is that ...
2
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1answer
34 views

Can we define ordinals such that the following sentences are independent of ZFC?

Can we explicitly define two ordinals $\alpha$ and $\beta$ in the language of $\{\in\}$ such that the following hold? ZFC proves that $\alpha$ and $\beta$ exist. ZFC proves that $\beth_\beta \neq ...
14
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1answer
299 views

Cardinality of a locally compact Hausdorff space without isolated points

I am interested in the following result: Theorem. A locally compact Hausdorff topological space $X$ without isolated points has at least cardinality $\mathfrak{c}$. To prove it, one can find two ...
4
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1answer
30 views

If $2^{\kappa}<\lambda$, how many subsets of size $\kappa$ are there of a set of size $\lambda$.

Assume both cardinals are infinite. Also assume AC as needed. So, the obvious bound is that there are no more than $\lambda^\kappa\leq 2^\lambda$ of them. But it seems there should be an easy bound ...
1
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1answer
24 views

Does for a set of cardinals a finite subset exist such that for any cardinal in the set a larger cardinal in the subset exists?

I am writing an essay for which I need to prove that sufficiently many graphs of a certain type exist. Is it true that for any set of sets (or set of cardinals) $S$ a countable subset $C$ exists such ...
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6answers
1k views

Why is the Continuum Hypothesis (not) true?

I'm making my way through Thomas W Hungerfords's seminal text "Abstract Algebra 2nd Edition w/ Sets, Logics and Categories" where he makes the statement that the Continuum Hypothesis (There does not ...
0
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2answers
36 views

A question about the size of the set of all countably-infinite subsets of a countably-infinite set

Let $A$ be a countably-infinite set , then how do we prove that the power set of $A$ and the set of all countably-infinite subsets of $A$ have the same cardinality (i.e. that there is a bijection) ? ...
2
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2answers
108 views

A property of strong limit cardinal

Suppose $\lambda$ is a strong limit cardinal, i.e. $\forall \alpha<\lambda \ 2^\alpha<\lambda$, and the cofinality of $\lambda$: $cf(\lambda)=\omega$. How do we show that $2^\lambda \leq ...
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2answers
85 views

Is there, for every set $X$, a set $Y$ for which $|Y| < |X|$ but $|\mathcal{P}(Y)| \geq |X|$?

As the title says, my question is: Is there, for every set $X$, a set $Y$ for which $|Y| < |X|$ but $|\mathcal{P}(Y)| \geq |X|$? I'm fairly certain this is true for finite sets but maybe ...
4
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4answers
212 views

Conclusion about cardinalty.

Assume that: $$\left| T \right| > {\aleph _0}$$ Why can't one assume immediately that: $$\left| T \right| \cdot \left| T \right| > \left| T \right| \cdot {\aleph _0}$$
4
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2answers
94 views

about the smallest $k$ that $V_k$ is a model of ZFC

Let $k$ to be the smallest ordinal that $V_k$ is a model of ZFC. I know that $k$ need not to be inaccessible cardinal,and $k$ has confinality $\omega$. Then how big is $k$? How to write down $k$ in ...
0
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1answer
35 views

What is the cardinality of the equivalence class

Consider this relation: $$R = \left\{ {\left\langle {f,g} \right\rangle \in {{\left\{ {0,1} \right\}}^N} \times {{\left\{ {0,1} \right\}}^N}|\exists k \in N\left| {\left\{ {i \in N|f(i) \ne g(i)} ...
0
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1answer
35 views

What is the cardinality of $M_2(\mathbb{R})$

What is the cardinality of $M_2(\mathbb{R})$, i.e the set of all 2 by 2 real matrices( $|M_2(\mathbb{R})|$)?
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3answers
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If the infinite cardinals aleph-null, aleph-two, etc. continue indefinitely, is there any meaning in the idea of aleph-aleph-null?

If the infinite cardinals aleph-null, aleph-two, etc. continue indefinitely, is there any meaning in the idea of aleph-aleph-null? Apologies if this isn't a sensible question, I really don't know too ...
2
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1answer
59 views

Suppose that S and T each have cardinality c. Show that $S\cup T $ also has cardinality c.

I tried to use the Cantor-Bernstein Theorem. First, we have $S\subset S\cup T$, so that $\left | S \right |\leqslant \left | S\cup T\right | $. This implies $\left | S\cup T \right |\geqslant c$. But ...
55
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6answers
3k views

Why is $\omega$ the smallest $\infty$?

I am comfortable with the different sizes of infinities and Cantor's "diagonal argument" to prove that the set of all subsets of an infinite set has cardinality strictly greater than the set itself. ...
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1answer
38 views

Cardinality of Orderings of $\mathbb{R}$

For a finite set $S$ there are $\vert S\vert!$ orderings of its elements. What is the cardinality of all orderings of $\mathbb{N}$? What would $$\vert \mathbb{N}\vert!$$ mean? Is it ...
0
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1answer
50 views

Is there always isomorphism between two sets that have the same cardinality?

Is there always isomorphism between two sets that has the same cardinality ? We only know that the two sets have the same cardinality. I tried to find a counter example but couldn't.
2
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5answers
341 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 ...
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0answers
13 views

Finding the cardinal of monotone increasing sequences of natural numbers [duplicate]

Find the cardinality of the set of all monotone increasing sequences of natural numbers. Well let's ignore the monotone increasing condition for a moment, then the cardinality of a set of all the ...
0
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4answers
30 views

Cardinality for all rational strongly increasing sequences

What is the cardinality for all rational strongly increasing sequences? Using diagonalization, I can show easily that for each list $f_n$ of sequnces, we can present a sequence which is not in ...
5
votes
3answers
214 views

Uncountable Cardinals without AC

I am doing an exercise, proving that without AC or Replacement that there are uncountable cardinals. As a point of reference I looked at the proof in Kunen's "The Foundations of Mathematics" that ...
5
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4answers
103 views

Cardinality of Irrational Numbers

I know and I have proved more than once that the set of irrational numbers ($\mathbb{I}$) is uncountable, but now I'm given to solve this problem: Show that $|\mathbb{I}|=|\mathbb{R}|$, How can I ...
0
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1answer
57 views

Cardinality Of Borel Sets

I was trying to show that Borel $\sigma$ algebra is smaller than lebesgue measurable sets. I could come up with a proof for the cardinality of lebesgue measurable sets being $2^c$. Cardinality of ...
4
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2answers
110 views

The regularity of successor cardinal

I was looking at two different proofs of the fact that successor cardinals are regular. It struck me as odd that both proofs used AC. Looking at the concepts involved in defining cofinality I feel as ...
6
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3answers
644 views

How do we prove the existence of uncountably many transcendental numbers?

I know how to prove the countability of sets using equivalence relations to other sets, but I'm not sure how to go about proving the uncountability of the transcendental numbers (i.e., numbers that ...