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.

learn more… | top users | synonyms (1)

0
votes
1answer
62 views

Can Aleph Numbers be multiplied?

i.e., does it make sense to say something like $(2 * \aleph_0) > \aleph_0$ ? The original question I was thinking about is: if A = $\mathbb{Z}$ and B = {the set of even integers} is it correct to ...
0
votes
1answer
26 views

What is the cardinality of a set of all finite subsets of $\Bbb{N}$? [duplicate]

I'm looking for cardinality of $P_{fin}(\Bbb{N})=\{x|x\subset\Bbb{N}$ and $x$ finite$\}$. I was told in my classes that it's $\aleph_0$, but how to prove it?
1
vote
1answer
33 views

Showing that the class of all sets of a particular cardinality is not a set.

How to show that the class of all sets of a particular cardinality ,say $h$ is not a set. My argument: I assume that I've shown the following lemma. Lemma: If $X$ is an infinite set of cardinality ...
0
votes
1answer
31 views

Cardinality of the set of all (real) discontinuous functions

This is a question from the book Introduction to Set Theory (Hrbacek and Jech), chapter 5, question 2.6. (Show that) The cardinality of the set of all discontinuous functions is ...
2
votes
1answer
29 views

Cardinality of sets: $|A|\le|B|\Rightarrow(|A\cup B|=|B|\land|A\times B|=|B|)$

My book of mathematical logic states the facts that, if we call $|X|$ the cardinality of set $X$, then, for any two sets $A,B$ such that $|A|\le|B|$, $$|A\cup B|=|B|\quad\text{ and }\quad|A\times ...
1
vote
1answer
25 views

Proving that if $A\leq_c B$ then $\chi(A)\leq_o \chi(B)$

By Hartog's Theorem we knoe that for every set $A$ There is a definite operation $\chi(A)$ which associates with each set $A$, a well ordered set $\chi(A) = (h(A),{\leq_{\chi(A)}}),$ such that $h(A) ...
0
votes
0answers
27 views

Connection beween closure property in forcing and preservation of $H_{\kappa}$

I would like to know about the relation between closure property of forcing notions and preservation of hierarchy of hereditary small sets, $\langle H_\lambda \mid \lambda\in \mathrm{Card}\rangle$, at ...
4
votes
2answers
147 views

Understanding proof of Hartogs’ Theorem on Set Theory

I'm trying to understand the proof of the Hartogs’ Theorem on page 100 of this book. My especific question is: If we have for each set $A$ $$\mathrm{WO}(A)=\{ (U,\leq_{U}) \, | \, U\subseteq A \, ...
1
vote
2answers
45 views

Injections from all ordinals into a set $X$

We are working in $\mathsf{ZF}$. Let $X$ be a set. Let $A$ be the class of all injections $f: \alpha \to X$ for arbitrary ordinals $\alpha$. I am quite sure that, in fact, $A$ is a set, since if ...
-1
votes
1answer
25 views

Suppose we have n independent nontrivial events. Prove: |Ω| ≥ 2^n . [closed]

Suppose we have n independent nontrivial events. Prove: $|\Omega| ≥ 2^n$ . By nontrivial events it means that $0 < P(E_i) < 1$, for $i=1,...,n$.
1
vote
0answers
22 views

Connectedness ( cardinality and connectedness) [duplicate]

$(X,d)$ metric space and $A\subset X$ and $A$ is connected. $$ \text{Card}(A) > 2 \implies \text{Card}(A) \geq \text{Card}(\mathbb{R}).$$ How do I prove it ?Waiting for your help?
0
votes
2answers
16 views

what is the cardinality of powerset of a union set?

Is there exist something like P(X+Y) (P STANDS FOR POWERSET)? I am confuse because power set is the set of all subset of Cartesian product, and X+Y wont give Cartesian product but (x,0) U (y,1), and ...
2
votes
1answer
18 views

Find the maximum number of a continuous function

Lets define a function $z:\mathbb{R}^\mathbb{R}\to\mathcal P(\mathbb R)$ that gives you the set of zeros of any $\mathbb R ^\mathbb R$ function. Now, we define a set $S=\{z(f):f\in\mathbb R ^\mathbb ...
3
votes
2answers
43 views

$\aleph_0 \aleph_1 =\aleph_1$? But I don’t know any way to prove or disprove it

What is the value of $\aleph_0 \aleph_1$? Clearly $\aleph_0\le \aleph_1$ implies $\aleph_0=\aleph_0\aleph_0\le \aleph_1 \aleph_0$ and again $\aleph_0 \aleph_1\le \aleph_1 \aleph_1=\aleph_1$. But ...
3
votes
1answer
36 views

I can't find the mistake in this argument (Cofinality and König's theorem)

I have some trouble explaining this apparent contradiction: we know that given $k>\aleph_0$ an infinite cardinal, $\mu=cof(k)$ is the minimum cardinal such that $k=\sum_{i\in \mu} k_i$ where ...
0
votes
1answer
29 views

If $\alpha$, $\beta$ are finite cardinals such that both are greater than $1$ and $\gamma$ is an infinite cardinal then $\alpha^\gamma=\beta^\gamma$.

Can someone suggest a rigorous proof of the following: If $\alpha$, $\beta$ are finite cardinals such that both are greater than $1$ and $\gamma$ is an infinite cardinal then $\alpha^\gamma = ...
-3
votes
0answers
42 views

The set of all cardinal number is bounded? [duplicate]

The set of all cardinal number is bounded? My assumption is unbounded. There must be some ordering in the cardinal number and also we have $0 < ℵ_1=2^{ℵ_0} <2^{2^ℵ0} <....$ there is no ...
3
votes
1answer
49 views

Cofinality of $2^{\aleph_\omega}$

Is the following statement correct: $\operatorname{cf}(2^{\aleph_\omega})=\aleph_0$? It appears in the "Jech" book. Wikipedia however states that $\operatorname{cf}(\aleph_\omega)=\aleph_0$. The ...
2
votes
1answer
49 views

Is it true that $({\aleph_1})^{\aleph_0} = {\aleph_1}$?

Is it true that $({\aleph_1})^{\aleph_0} = {\aleph_1}$? My scenario is as follows: The cardinal number of $\mathbb R$ is $|\mathbb R|={\aleph_1}$ and the cardinal number of the Cartesian product of ...
0
votes
1answer
12 views

closed unbounded set,regular cardinals,cofinality

Given two regular cardinals $\lambda>\mu$, why this set is a closed unbounded set in $\lambda$? {$\alpha$ | cf($\alpha$)=$\mu$ , $\alpha<\lambda$}
9
votes
1answer
101 views

Are injections harder to find than surjections?

Given two finite sets $A$ and $B$ with $|A|<|B|$ There are more functions from $B$ to $A$ than from $A$ to $B$ except when $|A|=1$ or $|A|=2,|B|=3,4$. See here for proof. It is also true there are ...
3
votes
3answers
353 views

Are there fields and ordered fields of every infinite cardinality?

On the top of my head, I cannot think of any fields of cardinality more than that of the reals. (It is known that the process of algebraic closure does not increase the cardinality of an infinite ...
0
votes
0answers
17 views

Mapping vectors to real numbers [duplicate]

Does there exist an invertible mapping that takes n-dimensional vectors (RN) to real numbers (R). Any countable set can be mapped in this way to another countable set. This is the sense in which ...
1
vote
1answer
53 views

Notation about cardinals

Does a cardinal $\mathcal{k}$ such that $2^\mathcal{k}=k^+$ have any special name? I never encountered any name for this property, but I think it is possible they have one.
0
votes
1answer
19 views

Prove that if A~B then Sym(A)~Sym(B).

I tried to prove it with sets. Really, truly clumsy. I know |A|=|B|. Can I simply conclude that |A|!=|B|! => Sym(A)~Sym(B)?? (Sym(A) for a set A is the set of all bijections from A to A.)
0
votes
0answers
24 views

Prove $α · β ≤ α · γ$ and $α^ β ≤ α^ γ$ for any three cardinals, where $ β ≤ γ$.

This is what I did: a. let $|A|=α, |B|=β, |C|=γ. |B|≤|C|$ and therefore there is an injection $f: B \to C$, such that $f(b)=c$ for some $b \in B, c \in C.$ Upgrading this function to ...
0
votes
3answers
77 views

Prove that $2^\aleph+\aleph$ equals $2^\aleph$.

How do I show this elegantly? I can't seem to find the right sets for it... Maybe there are some substantial laws I could use? I would appreciate your help... What I said is: $2^\mathfrak c$ is for ...
8
votes
0answers
96 views

Do an infinite set and its double have the same cardinality? [duplicate]

My question was inspired by this answer. Suppose $A$ is an infinite set. Does its double, $A\times\{0,1\}$, always have the same cardinality? In my head I quickly spotted a simple proof that the ...
0
votes
1answer
17 views

Transfinite fixed points of a function

Let the function $F\colon On \rightarrow On$ be defined by the following recursion: $F(0) = \aleph_0$ $F(\alpha+1) = 2^{F(\alpha)}$ (cardinal exponentiation) $F(\lambda) = \sup\{F(\alpha): \alpha ...
0
votes
4answers
25 views

Why we can conclude immediately that $x \in B$, if $(x, y) \in A \times B = B \times A$

The following statements are part of a proof involving cartesian products, specifically involving this theorem: $A \times B = B \times A \iff$ either $A = \emptyset$, $B= \emptyset$, or $A = B$ ...
0
votes
2answers
45 views

Showing the set of functions $\{0, 1\} \to \mathbb{N}$ is countably infinite.

I'm doing a question it asked me to show that $\mathbb{N} \times \mathbb{N}$ was countably infinite but I am stuck on the following part of the question: deduce that the set of all functions $f : ...
0
votes
3answers
50 views

Power of sets - $\{0,1\}^\mathbb{N} \simeq \mathbb{N}^\mathbb{N}$

I've got a problem with prove about cardinality of sets. How can I prove that $\lbrace 0,1 \rbrace^\mathbb{N} \simeq \mathbb{N}^\mathbb{N}$?
1
vote
1answer
69 views

Does $\aleph_0\cdot\kappa=\kappa$ for every $\kappa\ge\aleph_0$ hold in ZF?

It is easy to show that for any (Dedekind) infinite cardinal $\kappa$ we have $\aleph_0+\kappa=\kappa$. Definition of an infinite cardinal is a cardinal such that $\aleph_0\le\kappa$. (I believe ...
1
vote
2answers
52 views

Cardinality of the countably infinite product of a two-point set $\{0,1\}$?

I'm so confused about cardinalities of some sets. What is the countable infinite product of a two points set $\{0,1\}$? Does it have the same cardinality as the real number $\mathbb R$? Or is the ...
4
votes
1answer
64 views

Prove that $\forall\alpha\geq\omega$, $|L_\alpha|=|\alpha|$ without AC

Without using Axiom of Choice, prove that $$\forall\alpha\geq\omega,~~|L_\alpha|=|\alpha|,$$ in which $\alpha$ is an ordinals, $\omega$ is the set of natural numbers, $L_\alpha$ is the $\alpha$-th ...
1
vote
3answers
58 views

How to show that if $P\subseteq Q$ are finite sets, and $\#P=\#Q$, then $P=Q$?

Let $P, Q$ two finite sets such that : $$P \subset Q$$ $$\#P = \#Q$$ How do you show that $P = Q$ ? I don't see how I can show that $Q \subset P$
1
vote
3answers
81 views

Prove that R and P(R) does not have the same Cardinality.

How can I show such a thing in an elegant, valid way? I know it shouldn't be hard. Well it does have to be plausible and mathematically logic, but I guess I am not expected to rediscover the great, ...
0
votes
0answers
48 views

What is an undefinable ordinal? How large are the least undefinable ordinal and supremum of all undefinable ordinals?

What does it mean to say that a particular ordinal/cardinal number is "definable"? Where and how do we define this "definability"? If there are undefinable ordinals/cardinals, how large are the least ...
1
vote
1answer
27 views

Are normed spaces isodyne?

In general, do all non-empty open subsets of a normed space necessarily have the same cardinality?
3
votes
2answers
37 views

Cardinal Arithmetic proof issues.

Let $X$ be a finite set and let $x$ be an object which is not an element of $X$. Then $X \cup \{x\}$ is finite and $|X \cup \{x\}| = |X| + 1$. Proof. Let X be a finite set with cardinality n, ...
1
vote
1answer
37 views

What is the cardinality of the class of $0$ in $\mathbb{R}$?

What is the cardinality of the class of $0$ in $\mathbb{R}$? In other words: what is the cardinality of the class of all rational Cauchy sequences that converge to $0$?
2
votes
2answers
65 views

How are some infinities larger than other infinities

I heard an expressions, some infinities are larger than others recently, and they stated that it was proved to be so. I haven't been able to find this proof, and ...
0
votes
1answer
33 views

Prove that if $\: \forall_{t \in \mathbb{R}} \: \overline{\overline{A}}_t = c$ then $\overline{\overline{\bigcup_{t \in \mathbb{R}} \: A_t}} = c$

Prove that if $\: \forall_{t \in \mathbb{R}} \: \overline{\overline{A}}_t$ is equal to cardinality $c$, then $\:\overline{\overline{\bigcup_{t \in \mathbb{R}} \: A_t}}$ is also equal to cardinality ...
1
vote
1answer
20 views

Equalities of cardinal numbers

I need prove that: $2^{\aleph_{0}}=n^{\aleph_{0}}=\aleph_{0}^{\aleph_{0}}=c^{\aleph_{0}}=c$, for $n\geq2$. Where $c$ is the continuum. I know that $2^{\aleph_{0}}\leq ...
2
votes
1answer
45 views

Notation for the class of all cardinals

I have seen the notation for the class of all ordinals to be $\rm Ord$ or $\rm On$, is there an analogous notation for the class of all cardinals?
0
votes
1answer
38 views

If $2^{\text{cf } \kappa} < \kappa$, then $\kappa^+$ is the least possible value of $\kappa^{\text{cf } \kappa}$

I would like to show that if $2^{\text{cf } \kappa} < \kappa$, then $\kappa^+$ is the least possible value of $\kappa^{\text{cf } \kappa}$ where $\kappa$ is an infinite cardinal. I'm certain it ...
2
votes
0answers
47 views

Weight of topology related to sizes of open sets?

I'm wondering if there is a notion relating the weight of a topological space (=minimal base cardinality) to the sizes of its open sets. In particular I'm looking for properties of spaces, whose ...
2
votes
2answers
47 views

Does ${(x,y)\in\mathbb{R}\times\mathbb{R}:x,y\in\mathbb{Z}}$ have cardinality $\aleph_0$ or $c$?

So here's my intuition. Letting $S=\{(x,y)\in\mathbb{R}\times\mathbb{R}:x,y\in\mathbb{Z}\}$, $\bar S=\aleph_0$ because $(x,y)\in\mathbb{Z}\times\mathbb{Z}$, which can be shown to be equivalent to ...
4
votes
1answer
85 views

$H(\kappa)$ a model of all the axioms of ZFC for $\kappa$ not inaccessible

Is this possible? For each cardinal $\kappa$, we can define $H(\kappa) = \{ x \mid |trclx| < \kappa\}$, where trcl is the transitive closure, and $|A|$ is the cardinality of $A$. For every regular ...
3
votes
1answer
49 views

Cardinality of a set of functions from $\mathbb N$ to a set of real numbers

Let $S$ be the set of functions from $\mathbb N$ to the set $\{\sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{7}\}$. Determine $|S|$. I understand how to prove the converse case where $T$ is the set of ...