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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31 views

What is the cardinality of the set of all bijections from a countable set to another countable set? [duplicate]

What is the cardinality of the set of all bijections from a countable set to another countable set?
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1answer
24 views

Cardinality of the set difference.

If $B\subset A$, is it true that $Card(A)\backslash Card(B)$ is idempotent to $A\backslash B$ ? It seems to be true though I do not know how to prove it.
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1answer
49 views

Proving Cardinality of Sets

Prove that the sets E = {x ∈ ℕ : x = 2k, k ∈ ℕ } and ℕ have the same cardinality. A clear definition of cardinality was not given in this situation, so I understand cardinality to be (more or less) ...
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0answers
37 views

Is one-to-one correspondence required for determining cardinality?

If I wanted to show that two sets had the same cardinality, would I have to show that they are a function in which the two sets are in one-to-one correspondence? For example, if I wanted to start a ...
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1answer
80 views

Proving $2^{\aleph_0} = {\aleph_0}!$ [duplicate]

How can I show the existence of a injection $\phi:\{x|x \subset \omega \} \rightarrow \{f|f:\omega \rightarrow \omega$ is bijective$\} $?
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17 views

Will κ1,κ2,m cardinals. Given κ1≤κ2. prove: κ1⋅m≤κ2⋅m. [duplicate]

Will κ1,κ2,m cardinals. Given κ1≤κ2. prove: κ1⋅m≤κ2⋅m. Hi, I would be happy if someone could help me with this.. What I did until now:I replaced the cardinals with sets: |K1|=k1, |K2|=k2, |M|=m. From ...
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1answer
27 views

Confused by how to proof some statements about cardinals

I have a set of statements such as: Proof $\aleph_0+\aleph_0=\aleph_0$ I know that $|\Bbb Z|=\aleph_0$ and that for countable $A,B$ $A\cap B=\emptyset$: $|A\cup B|=|A|+|B|$. To this I add that ...
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1answer
39 views

Find the cardinality of $S=\{(x,y) \in \Bbb R^2 : 2x+3y<5\}$.

Find the cardinality of $S=\{(x,y) \in \Bbb R^2 : 2x+3y<5\}$. Attempt: I graphed this set, and I noticed that the simpler set $(0,1)^2=B\subset S$, and I thought these two sets had the same ...
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0answers
21 views

The size of the set of continuous function of periode T

I have a naive question. The Fourier series give an injection between continuous function of periode $T$ and the set of real valued sequences. But, don't we expect the set of continuous function of ...
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0answers
22 views

set difference between $\mathbb{R}$ and a countable set [duplicate]

How does one show that for a countable set $S$: $|\mathbb{R}\setminus S|=|\mathbb{R}|$ (I'm not familiar with the axiom of choise)
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0answers
35 views

Filter,dual ideal,definition of $\liminf_I \lambda$

We have this http://shelah.logic.at/files/506.pdf definition of $\lim \inf_I \bar{\lambda}$ in 1.1(3): for $I$ a filter on $\kappa$ let $I^+=2^\kappa \setminus I$.$$\lim \inf_I ...
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0answers
17 views

Cardinality relation between reducible sets

Suppose we are considering natural numbers, set $A$ and $B$ are two subsets of the natural number set, suppose set $A$ is many-one reducible to set $B$, i.e. there is a total computable function $f$ ...
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0answers
32 views

finding an injective function to prove cardinality equality

As part of a HW assignment in the course elementary set theory, I was given the following question: Prove that the set of all binary sequences (sequences of $0$ and $1$) except for the binary ...
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0answers
20 views

A question on product space [duplicate]

If $|X|=\mathfrak c$, then what is the cardinality of the product space $X^{\omega}$? Thanks very much.
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1answer
29 views

Cardinality and Set Thoery

(|x| = cardinal # of x for clarification) let A,B be two finite sets, show that $|A \cup B| = |A| + |B| -|A\cap B|$ Proof: let $x\in A\cup B$ $x \in (A -A\cap B) + (B- A\cap B) + (A \cap B)$ let ...
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0answers
47 views

Show that same cardinal number defines and equivalence relation

Same cardinal number must satisfy all three properties : symmetry, reflexivity, transitivity Symmetry Suppose bijection $f:A→A$, Then by definition, |A| = |A| Reflexivity Suppose bijections ...
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1answer
41 views

uncountable repetitions

I have a question (or two) about recursive naming conventions. Consider the following recursive naming sequence: Base step: Let S be any nonempty set. Let x be any arbitrary element of S. Let S* be ...
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0answers
24 views

What is the Cardinality of all symmetric density function pairs on reals?

$X=$(total number of all pairs of probability density functions $(f_0,f_1)$ on the real numbers) and let $Y=$(total number of all symmetric probability density functions $(f_0,f_1)$ on the real ...
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0answers
53 views

Finite sum over uncountable set

Consider the sum $S=\sum_{x\in I}P(x)$, where $P(x)$ are positive real numbers. When the index set $I$ is finite, $S$ is of course finite. When $I$ is countably infinite, it is also possible that $S$ ...
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1answer
41 views

$\operatorname{Card}(X) \leq\operatorname{Card}(Y)$ iff $\aleph (X) \leq \aleph(Y)$

For any two sets $X$ and $Y$, we write $\operatorname{Card}(X)\leq\operatorname{Card}(Y)$ if an injection $X \rightarrow Y$ exist. I have tried Suppose $\aleph (X) \leq \aleph(Y)$, where $\aleph ...
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0answers
27 views

determine the cardinality of $\{C \subseteq \mathbb N \space|\space \mathbb N - C \text{ is finite}\}$ [duplicate]

what is the cardinality of this set : $\{C \subseteq \mathbb N \mid \mathbb N - C \text{ is finite }\}$ So it must mean that $C$ is infinite, but even though its infinite we know how ...
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0answers
69 views

Vector spaces over Z/2Z,countably many predicates added,invariant

Suppose we have a class of vector spaces over $Z/2Z$ with predicates $P_n,n\in N$ added for independent subspaces of co-dimension 2. (BTW, independent in which sense,$P_n$ is independent with ...
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40 views

Is it possible to create a bijection between all pairs of reals and a real? [duplicate]

The title basically says it all. Is it possible to associate with each pair of reals, another unique real? I guess you could say I'm looking for functions of two real arguments that return a ...
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0answers
46 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 ...
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2answers
49 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 ...
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1answer
22 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($\alpha$)=$\mu$ , $\alpha<\lambda$}
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0answers
37 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 ...
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0answers
91 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 ...
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1answer
43 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 ...
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1answer
81 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 ...
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1answer
59 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 ...
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0answers
28 views

Cardinality of a set of injections [duplicate]

Let $A$ be the set of all injections $f: \mathbb{Z}_+ \rightarrow \mathbb{Z}_+$ What can we say about the cardinality of $A$ with respect to the cardinalities of $\mathbb{Z}_+$ and ...
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0answers
103 views

Existence of a Banach space with algebraic dimension $\alpha \neq \aleph_0$

For any given cardinal number $\alpha \neq \aleph_0$, is there a Banach space $X$ with algebraic dimension $\alpha$?
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0answers
27 views

$a\le b$ iff there exist $\left|A\right|=a, \left|B\right|=b$ and $A\subseteq B$ [duplicate]

$a\le b$ iff there exist $\left|A\right|=a, \left|B\right|=b$ and $A\subseteq B$ My Proof: $(\Leftarrow)$ $A\subseteq B$ implies immediately that $\left|A\right|\le\left|B\right|$. Hence, $a\le ...
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2answers
168 views

How can you compare the number of real numbers in the interval [0,1] and [0,10]?

There are infinite number of real numbers between 0 and 1,i.e in the interval [0,1]. So definitely there should be more numbers in the interval [0,10] because it includes the numbers in the first case ...
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0answers
24 views

Collection of sets with a given cardinality $\kappa$ is not set [duplicate]

Show that collection of all sets with cardinality $\kappa\neq0$, is not set. I'll state my approach and I need to see whether this idea is precise/precisable or not : First let $K$ be the set ...
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1answer
27 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 ...
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1answer
62 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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0answers
39 views

cardinality of finite subset

I have the following question: Prove that a set A has the same cardinality of a subset of a Set B, if and only if exists an injective function A to B. I find it hard to prove it because I can easily ...
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1answer
70 views

A seeming absurdity [duplicate]

I'm having a hard time getting over the following question, which appears in Schimmerling's "A Course on Set Theory." (Problem) Given that $\kappa$ and $\lambda$ are infinite cardinals with ...
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1answer
60 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 ...
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61 views

Can we conclude that $|\prod_{\xi \in Ord, \xi=1}^{\xi<\alpha}\xi|=2^{|\alpha|}$ for all infinite ordinal $\alpha$ in $\mathsf{ZFC}$?

According to this question, $\prod_{\kappa \in Crd, \kappa=1}^{\kappa<\aleph_\alpha}\kappa$, the cardinal product of all positive cardinals smaller than a given infinite cardinal $\aleph_\alpha$ ...
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0answers
42 views

What is the cardinality of the set of transfinite cardinals? [duplicate]

Possible Duplicate: Cardinality of all cardinalities What is the cardinality of the set of transfinite cardinals? The generalized continuum hypothesis ($2^{\aleph_a} = \aleph_{a+1}$) seems ...
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4answers
3k views

Why is the cardinality of irrational numbers greater than rational numbers?

This was asked by blogegog on a YouTube comment (gasp!): [Regarding Cantor's diagonal argument:] Couldn't I just make the same statement about rational numbers and say, 'take the largest ...
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3answers
347 views

Counterexample to cancellation law in cardinals addition

Charles C.Pinter - Set theory Let $a,b,c$ be any cardinal numbers. Give a counterexample to the rule: $$a+b = a+c \implies b=c$$ Does there exist a counterexample?
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1answer
42 views

If a countable union of sets has card $\mathfrak{c}$, prove at least one of them has card $\mathfrak{c}$ [duplicate]

If $A=\bigcup_{n=1}^{\infty}A_n$ and $A$ has cardinality $\mathfrak{c}$, where $\mathfrak{c}$ is the cardinal of the continuum, prove that at least one of the $A_n$ has cardinality $\mathfrak{c}$.
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3answers
43 views

I dont understand this sets question [closed]

Let $A = \{\{0\}\}$ and $B = \{0\}$. Which of the following statements are true and which are false? Justify each of your answers. $|A| = |B|$ (10 marks) $A \cap B = \emptyset $ (10 marks) $A \cap ...
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1answer
43 views

Set $T$ is Countably Infinite [closed]

How can it be shown that $$T = \{\,(i, j, k) \mid i, j, k \in\mathbb N\,\} $$ is countably infinite?
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3answers
200 views

The set of all functions from integers to a finite set is uncountable

Show that the set of functions from positive integers to the set $\{0,1,2,3,4,5,6,7,8,9\}$ is uncountable. I suspect I should use the diagonalisation argument but I'm not sure how to approach it. ...
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1answer
39 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.