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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1answer
43 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, ...
-2
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2answers
69 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
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2answers
52 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
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1answer
45 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
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1answer
50 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
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1answer
38 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
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1answer
60 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 ...
14
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0answers
141 views

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

I am trying to prove that there is no cardinal $\kappa$ such that $2^\kappa = \aleph_0$ . My attempt: We suppose it exists. Since $\kappa<2^\kappa$, in particular, $\kappa<\aleph_0$. But ...
4
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1answer
86 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
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2answers
110 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
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0answers
51 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
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0answers
58 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
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0answers
41 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
61 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 ...
9
votes
2answers
331 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
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1answer
70 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
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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 ...
2
votes
1answer
124 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 ...
5
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0answers
70 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} ...
2
votes
2answers
197 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
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3answers
49 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
48 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
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1answer
44 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 ...
2
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5answers
130 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
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0answers
52 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
73 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
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1answer
28 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
1answer
127 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
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3answers
201 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
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2answers
63 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
81 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
190 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
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1answer
44 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 $2^N =\{f|f:N\to\{0,1\} \text{ is a function} \}$?
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2answers
26 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
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1answer
31 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
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0answers
65 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
62 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 ...
1
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1answer
53 views

Cardinality question on vector spaces

Suppose $F$ is a field and $J$ an infinite set. Is it then true that $\mathrm{card} \ J<\mathrm{card} \ F^J$? ($F^J$ the set of maps $J\rightarrow F$) I know that $\leqslant $, but is it true and ...
1
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2answers
110 views

The cardinality of a union of two sets

Assume that the cardinality of the union of two sets is continuum. How to prove that at least one of the sets has the cardinality of a continuum? I suppose that it's possible to cope with it, using ...
1
vote
1answer
33 views

Cardinality of the power set $\mathcal P\left(S\right),$ where $S$ is a set of $15$ elements?

What is the cardinality of the power set $\mathcal P\left(S\right)$ where $S$ is a set of $15$ elements? I think the power set is a set of all the subsets of a given set or $2^n$. So would the ...
8
votes
2answers
600 views

Could someone explain aleph numbers?

I am having trouble understanding aleph numbers. I understand $\aleph_0$ is a countable infinity, but after that, I'm lost. What are $\aleph_1,\aleph_2,\aleph_3$, etc. to $\aleph_n$? Is there an ...
2
votes
2answers
73 views

Cardinality of the set of all numbers that modern math can define?

I have recently learned that the algebraic numbers are countably infinite, and that very few transcendental numbers are known. Are enough transcendentals known to make up an uncountable set, or is ...
1
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1answer
39 views

Prove that $\omega_{\alpha+1}$ is regular

Here, under the section 'Regular and Singular Cardinal', there is this sentence 'Assuming the Axiom of Choice, $\omega_{\alpha+1}$ is regular for each $\alpha$' . May I know how to prove this? Also, ...
0
votes
1answer
32 views

Haudorff Formula Set Theory

For every $\alpha$ and every $\beta$, $$\aleph_{\alpha+1}^{\aleph_{\beta}}=\aleph_{\alpha}^{\aleph_{\beta}} \cdot \aleph_{\alpha+1}$$ Proof: If $\beta \geq \alpha+1$, then ...
4
votes
3answers
173 views

Is there an infinite countable $\sigma$-algebra on an uncountable set

Let $\Omega$ be a set. If $\Omega$ is finite, then any $\sigma$-algebra on $\Omega$ is finite. If $\Omega$ is infinite and countable, a $\sigma$-algebra on $\Omega$ cannot be infinite and ...
2
votes
1answer
79 views

Cardinality for Kleene star and infinite Cartesian products.

Let $X$ be a finite set with at least 2 elements. Then the set of all finite-length "strings", $$X^* = \bigcup_{L \in \mathbb{Z}^+} \prod_{i=1}^L X_i = \{ (x_1, \ldots, x_L) : L \in \mathbb{Z}^+ ...
0
votes
1answer
80 views

Infinite Cardinal Addition Without the Axiom of Choice

In the book 'Introduction to Set Theory' by Hrbacek and Jech, cardinal addition is defined as $$\sum_{i \in I}{\kappa_i}=\left|\bigcup_{i \in I}{A_i} \right|$$ where $|A_i|=k_i$ for all $i \in I$ ...
3
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3answers
70 views

$|A\times B|= \text{max}(|A|,|B|)$ for infinite sets

I am fairly sure, given examples $\Bbb{R}\times \Bbb{R},\Bbb{R}\times \Bbb{Q},\Bbb{Q}\times \Bbb{Q} $, that this is correct, but do not know how to prove it. In my cited examples the proof has ...
0
votes
1answer
42 views

Mapping between set cardinalities

Please help me prove the following equalities between set cardinalities by explicitly showing an appropriate mapping: $$\left | (0,1) \right |= \left | (1,+\infty ) \right |$$
1
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1answer
60 views

Comparing cardinalities

Why these two sets are equinumerous? $$[0,1]^\Bbb N\text{ and }\Bbb Q^\Bbb N$$ Here is my reason: The set of rational numbers $\Bbb Q$ is countably infinite. However, $[0, 1]$ is not countable and ...