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
108 views

Definition of a $\gamma$-filtered ordinal

This definition is part of Hovey's book on model categories: Let $\gamma$ be a cardinal. An ordinal $\alpha$ is $\gamma$-filtered if it is a limit ordinal and, if $A\subseteq \alpha$ and $|A|\leq ...
6
votes
1answer
393 views

Finding the cardinality of a set

I have to construct a few things before I get to my question: it's possible we don't need all of it, but I am stuck on the very last step so I should recap everything so far. Let $\kappa$ be a ...
2
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2answers
241 views

Is the function of the limit = the limit of the function when I'm talking about continuous functions of ordinals?

Let $f: \bf{Ord} \to \bf{Ord}$ be a continuous, weakly increasing function, and let $\langle \alpha_{\xi} \mid \xi < \gamma \rangle$ be an increasing sequence of ordinals. Is it true that ...
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2answers
152 views

Club in a singular cardinal.

Let $\kappa$ be a singular cardinal with $\operatorname{cf}\kappa = \lambda > \omega$. Let $C = \{ \alpha_{\zeta} \mid \zeta < \lambda \}$ be a strictly increasing continuous sequence of ...
2
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1answer
213 views

Why can I write a singular cardinal as the limit of an increasing, continuous sequence of cardinals.

Let $\kappa$ be a singular cardinal, such that $\operatorname{cf}\kappa = \lambda < \kappa$. Now because $\operatorname{cf}\kappa = \lambda$, then I can write down an increasing sequence of ...
3
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1answer
311 views

Ideals in the ring of endomorphisms of a vector space of uncountably infinite dimension.

I know that if $V$ is a vector space over a field $k,$ then $\operatorname{End}(V)$ has no non-trivial ideals if $\dim V<\infty;$ $\operatorname{End}(V)$ has exactly one non-trivial ideal if ...
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2answers
1k views

Does a injective function $f: A \to B$ and surjective function $g : A\to B$ imply a bijective function exists? [duplicate]

Possible Duplicate: Proof of a Cantor-Bernstein-like theorem If $A, B$ are sets and there exists an injective function $f : A \to B$ and a surjective function $g: A \to B$, does this imply ...
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4answers
574 views

What is the cardinal number of all strictly increasing sequences?

What is the cardinal number of all strictly increasing sequences? I was able to find an injection from $(0,1)$ into the set of all strictly increasing sequnces by defining $i(a)=(a,1,2,3,4,\dots)$, ...
5
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2answers
2k views

An infinite subset of a countable set is countable

In my book, it proves that an infinite subset of a coutnable set is countable. But not all the details are filled in, and I've tried to fill in all the details below. Could someone tell me if what I ...
1
vote
1answer
148 views

Cardinality and ordinary mathematical induction

How do I approach this problem using ordinary mathematical induction? Notation: If A and B are sets then we will say they are the same cardinality and write $A\approx B$ if there is a one-to-one and ...
14
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1answer
303 views

The group of permutations with almost all points fixed is a maximal normal subgroup of the symmetric group.

Let $X$ be an infinite set and let $\operatorname{Sym}(X)$ be the symmetric group of $X.$ Let $N$ denote the set of all permutations $\pi\in\operatorname{Sym}(X),$ such that the complement of the set ...
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2answers
280 views

Enumeration of real 'sequences', cardinality, Cantors diagonal argument.

Let us fix a well ordering of the real numbers then consider a 'list' of some subset of the real numbers (with at least two elements) - called A -, enumerated by the well ordering. Say our well ...
5
votes
1answer
371 views

Is $2^\infty$ uncountable and is cardinality a continuous function?

I apologize if the title seems too vague, but this is how I was asked the question. So one of my friends intended to write an infinite sum like $\displaystyle \sum_{i=1}^{\infty} a_{2^i}$ . However, ...
5
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3answers
376 views

A cardinal exponentiation question

Suppose $\kappa$ and $\lambda$ are infinite cardinals and that $\lambda$ is regular. Kunen states somewhere that this means that we have ...
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2answers
673 views

A question about cardinal arithmetics without the Axiom of Choice

Is multiplication of infinite cardinals defined in ZF without Choice?
-1
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1answer
99 views

What would be the consequence of restricting multiplication by Zero to only Finite Cardinals? [closed]

What would be the consequence of restricting multiplication by Zero to only Finite Cardinals? Would this lead to contradictions? How could it be achieved?
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2answers
515 views

Multiplying Infinite Cardinals (by Zero Specifically)

On the Wikipedia page on Cardinal Numbers, Cardinal Arithmetic including multiplication is defined. For finite cardinals there is multiplication by zero, but for infinite cardinals only defines ...
0
votes
1answer
148 views

The minimal cardinal set from a choice between a set and its complement

Given a set $A$ I would like to know if already exist a math definition for a Set $L$ being: $L$="the smaller cardinal set from a choice between a set and its complement" i.e a set with this ...
2
votes
1answer
207 views

How to prove that every infinite cardinal is equal to $\omega_\alpha$ for some $\alpha$?

How to prove that every infinite cardinal is equal to $\omega_\alpha$ for some $\alpha$ in Kunen's book, I 10.19? I will appreciate any help on this question. Thanks ahead.
5
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2answers
2k views

How to understand the regular cardinal? [closed]

How to understand the regular cardinal? Could someone give me some examples?
2
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1answer
1k views

Existence of a bijection $f:ω_1×ω_1→ω_1$ for the first uncountable ordinal $ω_1$?

May I ask why "Let $ω_1$ be the first uncountable ordinal. There's a bijection $f:ω_1×ω_1→ω_1$. " Thanks!
3
votes
2answers
102 views

If a line segment is divided into 2 parts then one of the parts is equinumerous to the original segment.

Assume we have a set $X$ which is a (closed) line segment. Prove that if we split $X$ into 2 parts $X_1$ and $X_2$ then at least one of those sets would have the same cardinality as $X$. My attempt: ...
6
votes
2answers
2k views

What is known about the power set of the real number line?

Cantor's theorem states that the cardinality of a set's powerset is strictly greater than that of the set itself. This clearly applies to the reals also; if I'm not mistaken, the cardinality of the ...
3
votes
4answers
2k views

Proving $\mathbb{N}^k$ is countable

Prove that $\mathbb{N}^k$ is countable for every $k \in \mathbb{N}$. I am told that we can go about this inductively. Let $P(n)$ be the statement: “$\mathbb{N}^n$ is countable” $\forall n \in ...
2
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1answer
249 views

Is every subset of $\{0,1\}^*$ countable?

The set of all subsets of $\{0,1\}^*$ is not countably infinite, but does this mean that every subset is countable?
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2answers
2k views

Proving the Cantor Pairing Function Bijective

How would you prove the Cantor Pairing Function bijective? I only know how to prove a bijection by showing (1) If $f(x) = f(y)$, then $x=y$ and (2) There exists an $x$ such that $f(x) = y$ How would ...
6
votes
1answer
128 views

Why do $2^{\lt\kappa}=\kappa$ and $\kappa^{\lt\kappa}=\kappa$ when the Generalized Continuum Hypothesis holds?

I'm going to assume GCH here. If that holds, then why do we have the equalities $2^{\lt\kappa}=\kappa$ for every $\kappa$, and $\kappa^{\lt\kappa}=\kappa$ for all regular $\kappa$?
11
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2answers
406 views

Uncountable chains

$P(\mathbb N)$ = power set of $\mathbb N$. $A \subset P(\mathbb N)$ is a chain if $a,b \in A \implies$ either $a \subseteq b$ or $ b \subseteq a$ That is we have something like this: $$\ldots a ...
15
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3answers
2k views

The Aleph numbers and infinity in calculus.

I have a fairly fundamental question. What is the difference between infinity as shown by the aleph numbers and the infinity we see in algebra and calculus? Are they interchangeable/transposable in ...
6
votes
2answers
154 views

What is the product of finitely indexed alephs?

I'm simply curious about why the following equality holds: $ \displaystyle\prod_{n\lt\omega}\aleph_n=\aleph_\omega^{\aleph_0}. $ Much thanks!
5
votes
1answer
135 views

Filter completeness question

I have a question about filters which I suspect has a very simple answer (hence my asking it here as opposed to MO): Let $F$ be a filter on an infinite set $X$. Then $F$ is "countably closed" if for ...
6
votes
2answers
200 views

What is the product of all nonzero, finite cardinals?

To be specific, why does the following equality hold? $$ \prod_{0\lt n\lt\omega}n=2^{\aleph_0} $$
6
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3answers
168 views

Cardinality of a some linear ordering is at most that of a given cardinal?

This is an intuitive idea that I've used for a while, but don't know how to explain formally. Suppose $(A,\prec)$ is some linear ordering, and each initial segment of $A$ has cardinality strictly ...
5
votes
1answer
424 views

Intuition about the size of $\aleph_k$ with $k>1$

Assuming CH for simplicity, I know of some more or less intuitive way to think about difference in sizes of $\aleph_0$ and $\aleph_1$. The most straightforward is the distinction of natural/rational ...
2
votes
2answers
2k views

How to Understand the Definition of Cardinal Exponentiation

I'm having trouble understanding the definition of cardinal exponentiation. Let's start with the definitions / claims I've been given: For any finite sets $A,B$, such that if $|A|=a$ and $|B|=b\neq ...
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2answers
560 views

axiom of choice: cardinality of general disjoint union

I have this exercise involving the axiom of choice, but I don't understand where it's needed: Let $(X_i)_{i \in I}$ and $(Y_i)_{i \in I}$ be pairwise disjoint sets with $|X_i| = |Y_i|$. Prove, using ...
3
votes
2answers
2k views

Union of Uncountably Many Uncountable Sets

I know that the union of countably many countable sets is countable. Is there an equivalent statement for uncountable sets, such as the union of uncountably many uncountable sets is uncountable? ...
0
votes
1answer
155 views

Cardinality of Set of Simple Closed Curves

What is the cardinality of the set of all simple closed curves in $R^2$? Furthermore, what resources are there which present a proof, if any, of said cardinality?
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4answers
717 views

Problems about Countability related to Function Spaces

Suppose we have the following sets, and determine whether they are countably infinite or uncountable . The set of all functions from $\mathbb{N}$ to $\mathbb{N}$. The set of all non-increasing ...
1
vote
1answer
137 views

Cardinality of set of normal functions $f \colon \omega_{\alpha} \to \omega_{\alpha}$

What is the cardinality of the set of all normal functions $f \colon \omega_{\alpha} \to \omega_{\alpha}$, where $\omega_{\alpha}$ is the initial ordinal of $\aleph_{\alpha}$?
8
votes
1answer
432 views

For every $n < \omega$, $\aleph_n^{\aleph_0} = \max(\aleph_n,\aleph_0^{\aleph_0})$

I have proved that if $\aleph_n \leq \aleph_0^{\aleph_0}$, then $\aleph_n^{\aleph_0} \leq \max(\aleph_n,\aleph_0^{\aleph_0})$. Clearly $\aleph_n \leq \aleph_n^{\aleph_0}$ and $\aleph_0^{\aleph_0} \leq ...
5
votes
3answers
478 views

How many $p$-adic numbers are there?

Let $\mathbb Q_p$ be $p$-adic numbers field. I know that the cardinal of $\mathbb Z_p$ (interger $p$-adic numbers) is continuum, and every $p$-adic number $x$ can be in form $x=p^nx^\prime$, where ...
4
votes
4answers
578 views

What is the cardinality of the group of bijections from $\Omega$ to $\Omega$ with finite support?

These questions cropped up in the discussion in this question, What is the cardinality of the group of bijections from $\Omega$ to $\Omega$ with finite support, where $\Omega=\mathbb{N}$? ...
4
votes
3answers
2k views

Cardinality and infinite sets: naturals, integers, rationals, bijections

I have alot of questions. Do Infinite sets have the same cardinality when there is a bijection between them? Are $\mathbb{N}$ and $\mathbb{Z}$ infinite sets? I assume they are, but why? Why does ...
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2answers
400 views

Implications of continuum hypothesis and consistency of ZFC

I've been a bit confused whilst doing some reading. I think the confusion arises because I am trying to read Wikipedia on topics without being able to work along. Anyway, I have read that, of course, ...
4
votes
1answer
74 views

$|S_X|=|S_Y| \Leftrightarrow |X|=|Y|$

Reading this problem I remembered trying to solve the following problem. For a set $A$, denote by $S_A=\{ f : A \to A | f \text{ is bijective }\}$. Denote by $|X|$ the cardinal number of $|X|$. ...
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votes
4answers
613 views

Fractional cardinalities of sets

Is there any extension of the usual notion of cardinalities of sets such that there is some sets with fractional cardinalities such as 5/2, ie a set with 2.5 elements, what would be an example of such ...
4
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4answers
2k views

Cardinality of the Irrationals [duplicate]

Possible Duplicate: Proof that the irrational numbers are uncountable We previously proved that $\mathbb{Q}$, the set of rational numbers, is countable and $\mathbb{R}$, the set of real ...
3
votes
2answers
228 views

How can I show that the set of reals and the set of pairs of reals have the same cardinality?

How can I show that the set of reals and the set of pairs of reals have the same cardinality? I know that since reals are uncountable infinite, I can't create a list of reals and talk about the ...
9
votes
2answers
2k views

Cartesian Product of Two Countable Sets is Countable

How can I prove that the Cartesian product of two countable sets is also countable?