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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3
votes
2answers
27 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
35 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
55 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
32 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
18 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
44 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
37 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
45 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
44 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
80 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
44 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 ...
3
votes
0answers
46 views

Is it possible to create division via Set Theory?

I've been reading a book on Set Theory (Charles C. Pinter), and it says, ...set theory is recognized to be the cornerstone of the "new" mathematics... [emph. added] and that ...we can still ...
1
vote
1answer
36 views

A question dealing with cardinals, and axiom of choice.

I am given sets $A$,$B$ such that there exists $f:A\rightarrow B$ s.t. $f$ is onto $B$. I am trying to show that $B\le A$ Let $b\in B$, consider $\{a\in A \mid f(a) = b\}$, assuming axiom of choice, ...
-1
votes
2answers
51 views

Partition of set of size more than $2^{\aleph _0}$ [closed]

Can every set of size more than $2^{\aleph _0}$ be partitioned into subsets, such that each is non-singleton and each has size at-most $2^{\aleph_0}$? Can every set of size more than $2^{\aleph _0}$ ...
-1
votes
2answers
40 views

Is the cardinality of the set of all isolated points in a second countable metric space at-most $\aleph_0$?

Is the cardinality of the set of all isolated points in a second countable metric space at-most $\aleph_0$ ?
0
votes
1answer
20 views

Cardinality of the set of all functions with finite support

Let F be a countable field and B an infinite set. Let $(F^B)_0$ be the set of all functions with finite support from F to B. Is it true that $|(F^B)_0|=|B|$?
1
vote
1answer
39 views

Arithmetics of cardinalities: if $A=C$ and $B=D$ then $A\times B=D\times C$

Suppose that $A, B, C$, and $D$ are sets with the cardinalities related as $A=C$ and $B=D$. Prove that the cardinality of $A\times B$ is equal to the cardinality of $D\times C$. I know that I must ...
0
votes
0answers
22 views

$\mathbb R$ ~ $[0,1) \cap (\mathbb R\ $\ $\mathbb Q)$

$\mathbb R$ is the set of the real numbers. $\mathbb Q$ the set of the rational numbers. So how I can prove this? $\mathbb R$ ~ $[0,1) \cap (\mathbb R\ $\ $\mathbb Q)$ I am also not sure what means ...
0
votes
1answer
27 views

A question about an exercise on basic cardinal arithmetic.

I just want to make sure that I have proved the following exercise correctly. Given two cardinal numbers $a$ and $b$ where $a$ is infinite. I was to show that $2\le b \le 2^a \implies b^a=2^a$ I ...
1
vote
1answer
26 views

cardinal numbers proof

Suppose $a,b$ are cardinals where $a$ is finite and $b$ is infinite. I want to prove that $b^a=b$. The book gives a hint saying to use repeated multiplication of cardinals to do it. I have proved ...
8
votes
0answers
41 views

There's no cardinal $\kappa$ such that $2^\kappa = \aleph_0$

I am trying to prove that a cardinal $\kappa$ such that $2^\kappa = \aleph_0$ . My attempt: We suppose it exists. Since $\kappa<2^\kappa$, in particular, $\kappa<\aleph_0$. But that implies ...
4
votes
1answer
73 views

Is the existence of such a transitive model $M$ of ZFC consistent?

Questions. Q0. Does anyone know of a refutation, in ZFC, of the following statement? Q1. If not, does ZFC plus large cardinals prove its consistency? Statement. There exists a transitive model $M$ ...
1
vote
2answers
88 views

Cardinality of set difference

How to prove this: given an infinite set $B$ and $A\subset B$ such that $|A| < |B|$, then $|B-A| = |B|$? Progress So, I do understand the definition of $|A| < |B|$ (it means $|A| \le |B|$ and ...
1
vote
0answers
40 views

Existence(?) of a set whose cardinality cannot be determined in ZFC

(First, I apologize if I display any fundamental misunderstanding of how set theory works.) I had a question regarding the limitations of ZFC (assuming its consistency, of course.) Is there any ...
1
vote
0answers
54 views

Is there an axiom for ZFC that totally orders Cichon's diagram without collapsing it?

Is there a known axiom for ZFC that: Fixes a total ordering on the cardinal numbers appearing in Cichon's diagram. Is "natural looking" - in particular, its not allowed to be a conjunction of ...
0
votes
0answers
36 views

Applications of Infinitary Matrices in Set Theory

Matrices have a natural generalization to infinitary context. There are few known applications of such matrices in set theory. For example one may use Ulam matrices to show that real-valued measurable ...
2
votes
1answer
29 views

Show that an infinite set $C$ is equipotent to its cartesian product $C\times C$

So, as the title says I'd like to give a proof of the fact that if $C$ is an infinite set then it is equipotent or equivalent to its cartesian product $C\times C$ using Zorn's Lemma (and of course ...
7
votes
1answer
173 views

Are there non-equivalent cardinal arithmetics?

‎Generalizing a concept in mathematics is always a problematic situation. In most cases there are several ways to generalize a notion and it is not easy to decide if a particular generalization is ...
0
votes
1answer
61 views

understanding cardinal numbers arithmetic

I have a question about notation in a book I'm reading on set theory and beside of my question I will be glad for a recommendation for a good book that explains well cardinal numbers arithmetic. If ...
19
votes
2answers
1k views

Are there number systems corresponding to higher cardinalities than the real numbers?

As most of you know, the set $\omega$ with cardinality $\aleph_0$ corresponds to what we normally know as the natural numbers $\mathbb{N}$, and the set $\mathcal{P}(\omega)$ with cardinality ...
1
vote
1answer
88 views

Is the set of all cardinals smaller then a strongly inaccessible cardinal closed?

Given a strongly inaccessible cardinal $k$ (i.e. $k$ is regular uncountable and for each $\lambda < k$, $2^\lambda < k$), is the set of all cardinals smaller then $k$ closed or open? Mahlo ...
4
votes
0answers
54 views

For cardinals, if $\mathfrak{a}\ne\mathfrak{b}$ then $2^\mathfrak{a}\ne 2^\mathfrak{b}$

In the usual ZF (or ZFC) set theory, let $\mathfrak{a}$ and $\mathfrak{b}$ be cardinal numbers. Is it correct that one can neither prove nor disprove the statement: $$\mathfrak{a}\ne\mathfrak{b} ...
3
votes
2answers
179 views

Cardinal number of the set of all one-to-one mappings of $A$ onto itself. [closed]

This is an exercise in Naive Set Theory by P. R. Halmos. If $\text{card }A=a$, what is the cardinal number of the set of all one-to-one mappings of $A$ onto itself? What is the cardinal number ...
0
votes
3answers
44 views

determining the cardinality [duplicate]

Let $S$ be the collection of closed intervals in the real line whose lengths are positive rational numbers. Determine the cardinality of $S$. Justify your answer As I understand, $S$ will be an ...
0
votes
1answer
38 views

Cardinality of a set of closed intervals

What is the cardinality of the set S of all closed intervals on the real number line with rational (positive) lengths? I believe the number of intervals with a specific fixed length but varying start ...
1
vote
1answer
42 views

a question on cardinality

Suppose $S$ and $T$ are sets such that $|S|=|T|$ Prove that $|\mathcal{P}(S)|=|\mathcal{P}(T)|$. To start with, $|S|<|\mathcal{P}(S)|$; $|T|<|\mathcal{P}(T)|$. Just the statement itself sounds ...
1
vote
5answers
93 views

Let A be a set of all infinite sequences consisting of 0's and 1's. Prove that A is not countable.

Sequences such as {010101010101...., 10100100100...., etc} if i am not wrong these sequences can represent all the real numbers in the binary format, so a such a set will not be countable. but i am ...
1
vote
0answers
48 views

Characterizability in $L^2_{\kappa^+\omega}$

I'm reading an article on second order characterizability. At some point in the article it proves that any model $A$ of cardinality $\kappa$ must be characterizable in $L^2_{\kappa^+\omega}$. I.e. ...
2
votes
2answers
61 views

A forcing that is $\omega_1$-closed and $\omega_2$-c.c.

I am reading an article (on second order characterizability) which at some point in a proof states that by forcing with $\mathbb P=\{f:\alpha\to\{0,1\},\alpha\in\omega_1\}$ we do not add subsets to ...
0
votes
1answer
24 views

a question concerning multiplication of cardinal numbers

Consider $\{B_i\}$ where $i\in I$ and $I$ is countable infinite. $|B_i|=|B_j|=n$ for all $i,j$ and $n \ge |\mathbb{N}|$. I want to show that $| \large \cup_{i\in I}$$B_i|=n$ I am given that $a*a=a$ ...
2
votes
0answers
90 views

How large is an uncountable regular cardinal which is closed under arbitrary fast operators?

Let $Card$ be the proper class of all cardinals, define an infinite set of operators like $\otimes_{n}:(Card\setminus \omega)\times (Card\setminus\{0\})\longrightarrow Card$ which are defined for each ...
2
votes
3answers
124 views

Example of set of cardinality $\aleph_2$

I am looking for an example of a set of cardinality $\aleph_2$, such as the continuum is an example for cardinality $\aleph_1$.
1
vote
2answers
51 views

Showing $\prod_{n < \omega} n = 2^{\aleph_0}$ [duplicate]

I have to show that $\prod_{n < \omega} n = 2^{\aleph_0}$. I'm having trouble getting started. I know that $2^{\aleph_0}$ is the set of binary sequences, or the space of functions from $\mathbb{N}$ ...
3
votes
1answer
70 views

$\kappa$ ineffable $\Rightarrow$ $\kappa$ tree-property

Let $\kappa$ be an uncountable, regular cardinal. We call $\kappa$ ineffable iff for every sequence $(A_\xi \colon \xi < \kappa)$ of subsets $A_\xi \subseteq \xi$ there is a stationary subset $S ...
3
votes
3answers
172 views

The Free Set Lemma

The statement of the lemma is as follows: if $$f: \omega_1 \rightarrow \{x\ :\ x\ \textrm{is finite}\}$$ then there is an uncountable $S \subseteq \omega_1$ such that for all distinct $\alpha,\ \beta ...
0
votes
1answer
35 views

How to define an explicit bijection from P(N) to 2^N [closed]

How do I define an explicit bijection between the power set of N and $2^N$ with $2^N =\{f|f:N\to\{0,1\} \text{ is a function} \}$?
0
votes
2answers
23 views

How do I prove an equivalence of these two statements about Cantor's hypotheses?

How do I prove that the continuum hypotheses as stated by George Cantor (There are no sets with cardinality between the cardinality of the real and the cardinality of the rational numbers) is ...
0
votes
1answer
24 views

smallest cardinal greater than an infinite ordinal is a regular cardinal

let $\alpha$ be an infinite ordinal, and $\alpha^+$ be the smallest cardinal greater than $\alpha$. Show that $\kappa^+$ is a regular cardinal. This is for homework, but I'm not really sure where to ...
4
votes
0answers
58 views

$\mu$-clubs and stationary sets consisting of elements with cofinality $\mu$

Let $\mu < \kappa$ be infinite cardinals. A set $C$ is called a $\mu$-club in $\kappa$, if it is unbounded in $\kappa$ and contains all its limit points of cofinality $\mu$. Now let $T \subset S ...
3
votes
1answer
58 views

If $2^{\aleph_{\beta}}\geq \aleph_{\alpha}$, then $\aleph_{\alpha}^{\aleph_{\beta}}=2^{\aleph_{\beta}}$.

If $2^{\aleph_{\beta}}\geq \aleph_{\alpha}$, then $\aleph_{\alpha}^{\aleph_{\beta}}=2^{\aleph_{\beta}}$. Proof: Note that if $\beta \geq \alpha$, then we have ...