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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367 views

cardinality of infinite sets

prove or disprove: If two infinite sets $A$,$B$ have the same cardinality, then $A\cup B$ and $A$ have the same cardinality. I even cannot make a judgement. P.S: Can this be done without using ...
6
votes
1answer
374 views

What is the standard proof that $\dim(k^{\mathbb N})$ is uncountable?

This is my (silly) proof to a claim on top of p. 54 of Rotman's "Homological algebra". For $k$ an infinite field (the finite case is trivial) prove that $k^\mathbb{N}$, the $k$-space of functions ...
6
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0answers
232 views

About singular $\beth_{\alpha}$ for limit ordinals $\alpha$

Why are there cofinitely many regular cardinals of type $\beth_{\beta}$ below any given singular $\beth_{\alpha}$, if $\alpha$ is a limit ordinal?
6
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0answers
131 views

Is there an upper bound to the number of rings that can be obtained from a semigroup with zero by defining an additive operation?

Let $\mathscr S$ be the class of all semigroups with zero. For $(S,\times,0)\in\mathscr S,$ I want to count additive operations $+$ on $S$ such that $(S,+,\times,0)$ is a ring (possibly without ...
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votes
3answers
765 views

Prove that transcendental numbers exist: Are there less paniful ways of doing it?

I've found this exercise on Boolos' Logic and Computability: A real number $x$ is called algebraic if it is a solution to some equation of the form: $$c_{\small d}x^{\small ...
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5answers
2k views

Show that open segment $(a,b)$, close segment $[a,b]$ have the same cardinality as $\mathbb{R}$

a) Show that any open segment $(a,b)$ with $a<b$ has the same cardinality as $\mathbb{R}$. b) Show that any closed segment $[a,b]$ with $a<b$ has the same cardinality as $\mathbb{R}$. ...
5
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2answers
554 views

Why are infinite cardinals limit ordinals?

My book states this as obvious, but it isn't so trivial to me. thanks
5
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3answers
128 views

$|\mathbb{R}^\mathbb{R}|$ vs $|P(\mathbb{R})|$ Which is bigger?

$|\mathbb{R}^\mathbb{R}|$ vs $|P(\mathbb{R})|$ where $\mathbb{R}^\mathbb{R} =\{f | f:\mathbb{R} \rightarrow \mathbb{R}\}$ Are they equal? Which is bigger? How can I prove it?
5
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3answers
271 views

Question about Generalized Continuum Hypothesis

I wonder how the Generalized Continuum Hypothesis reveal that $A\times A$ is equivalent to $A$? $A$ is any infinite set.
5
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3answers
421 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 ...
5
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2answers
486 views

In set theory, what does the symbol $\mathfrak d$ mean?

What's meaning of this symbol in set theory as following, which seems like $b$? I know the symbol such as $\omega$, $\omega_1$, and so on, however, what does it denote in the lemma? Thanks ...
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2answers
2k views

Cantor-Bernstein-like theorem: If $f\colon A\to B$ is injection and $g\colon A\to B$ is surjective, can we prove there is a bijection as well?

I've been trying to find this proof: If there exists $f \colon A\to B$ injective and $g \colon A \to B$ surjective, prove there exists $h \colon A \to B$ bijective. I thought of using ...
5
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3answers
462 views

Infinite sets with cardinality less than the natural numbers

Are there any infinite sets that have a lower cardinality than the natural numbers? Is there a proof of this?
5
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1answer
539 views

Cofinality of cardinals

I have read some of the related questions (cofinality and its consequences, for example) about cofinality, but I still have no idea how to calculate the cofinality of an aleph. So, let's say I have ...
5
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3answers
218 views

Question on a proof from Jech's Set Theory of: If $X$ is a set of cardinals, then $\sup X$ is a cardinal.

The proof: Let $\alpha= \sup X$. If $f$ is a bijective mapping of $\alpha$ onto some $\beta < \alpha$, let $\kappa \in X$ be such that $\beta < \kappa \le \alpha$. Then $|\kappa|=|\{f(\xi): \xi ...
5
votes
3answers
300 views

What does $H(\kappa)$ mean?

As the typical references (Wikipedia, Mathworld, etc.) don't seem to address this satisfactorily, I figured this would be a good place to put a nice formal definition. Hence: I've heard that if ...
5
votes
1answer
117 views

Is $\alpha^{\beta}=2^{\beta}$ for all infinite cardinalities $\alpha, \beta$?

Is $\alpha^{\beta}=2^{\beta}$ for all infinite cardinalities $\alpha, \beta$? I was wondering about this since I've encountered examples of times where this holds, but I can't seem to prove it myself ...
5
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2answers
2k views

Cardinality of $\mathbb{R}$ and $\mathbb{R}^2$

I am working on this exercise for an introductory Real Analysis course: Show that |$\mathbb{R}$| = |$\mathbb{R}^2$|. I know that $\mathbb{R}$ is uncountable. I also know that two sets $A$ and ...
5
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4answers
1k views

What are Aleph numbers intuitively?

I cannot get my head around the concept of the `types' of Aleph infinity. What is an easy intuitive way to see when you are given the integer numbers $\aleph_0$ the $\aleph_1$ will follow?
5
votes
1answer
147 views

What is the value of $\aleph_1^{\aleph_0}$?

Is there any neat way to calculate the value of $\aleph_1^{\aleph_0}$?
5
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3answers
309 views

The cardinality of a countable union of sets with less than continuum cardiality

Can continuum be a countable union of sets with cardinality, less than continuum? I can prove it for a finite union by mathematical induction from this: $\mathbb R = A_1 + A_2 => |\mathbb R| = ...
5
votes
1answer
390 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 ...
5
votes
4answers
605 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 ...
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3answers
569 views

Cardinality != Density?

I was in a discussion where I argued that the density of two sets of the same cardinality could be different in respect to the infinite range of non-negative integers. Does cardinality imply that any ...
5
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4answers
399 views

Cardinality of Irrational Numbers

I know and I have proved more than once that the set of irrational numbers ($\mathbb{I}$) is uncountable, but now I'm given to solve this problem: Show that $|\mathbb{I}|=|\mathbb{R}|$, How can I ...
5
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3answers
243 views

Uncountable Cardinals without AC

I am doing an exercise, proving that without AC or Replacement that there are uncountable cardinals. As a point of reference I looked at the proof in Kunen's "The Foundations of Mathematics" that ...
5
votes
2answers
141 views

What is the value of $\beth_{\omega_1}^{\aleph_0}$?

It is well known that $\beth_\omega^{\aleph_0} = \beth_{\omega+1}$. This follows since for strong limit $\kappa$, we have $\kappa^\kappa = \kappa^{\mathrm{cf}(\kappa)}.$ Question. To the extent that ...
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5answers
168 views

$\mathrm{card} ( \mathbb{Q})=\mathrm{card}( \mathbb{Q^c})$: Overcoming Wrong Intuition

This is a widespread intuitive argument, asserting that $\mathrm{card} ( \mathbb{Q})=\mathrm{card}( \mathbb{Q^c})$: Between any two rational numbers there's an irrational one and vice versa. So ...
5
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2answers
478 views

Cardinal Arithmetic versus Ordinal Arithmetic

I am reading Philosophy, not Set Theory, so please excuse the naivety of my question. My question concerns the wildly different character of ordinal arithmetic versus cardinal arithmetic. The ...
5
votes
1answer
195 views

Why continuum function isn't strictly increasing?

Is there any example that for cardinal numbers $\kappa < \lambda$, we have $2^\kappa = 2^\lambda$? My guess is that it only depends on whether GCH holds. Is it true?
5
votes
1answer
93 views

Landing between $\beth_\lambda$ and $\beth_{\lambda+1}.$

Main Question. Is it consistent with ZFC that there exists a limit ordinal $\lambda$ and a cardinal number $\nu$ satisfying $$\beth_\lambda < \beth_\lambda^\nu < \beth_{\lambda+1}?$$ I am also ...
5
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1answer
106 views

Existence of a regular uncountable $\aleph_{\alpha}$ without $\mathsf{AC}$

Set theory (Jech) $\text{p.}\;27:$ It is an open problem whether one can prove without the axiom of choice that there exists a regular uncountable $\aleph_{\alpha}\;($the informed guess is that ...
5
votes
1answer
147 views

Injective or constant function on measurable cardinal

I'm working on the following question: Let $\kappa$ regular cardinal, and $f:\kappa \rightarrow \kappa$. Prove that $\{\alpha < \kappa\mid f|_\alpha : \alpha \rightarrow \alpha\}$ is a ...
5
votes
1answer
132 views

Aleph arithmetic question

We want to prove that: $$\aleph_2^{\aleph_0} = \aleph_2\aleph_1^{\aleph_0}$$ My idea was to approach this by doing a Schroder-Bernstein style argument and proving this by showing two inequalities, ...
5
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2answers
119 views

Reference request, self study

I'm looking for references (books/lecture notes) for : Cardinality without choice, Scott's trick; Cardinal arithmetic without choice. Any suggestions ? Thanks in advance.
5
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1answer
83 views

$ZFC^- + AFA$ and infinite cardinals

$ZFC^-+AFA$ is a non-well-founded set theory, where $ZFC^-=ZFC-FA$ is $ZFC$ without the axiom of foundation, and $AFA$ is an anti-foundational axiom With the axiom of foundation we have that every ...
5
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1answer
75 views

Cardinal Arithmetic: $\beth_\omega$

Is the following equality independent of ZFC: $$\bigcup_{\aleph_{\beta}<\beth_{\omega}}2^{\aleph_{\beta}}=\beth_\omega $$ Consistent: Now that the equality is consistent with ZFC since it holds ...
5
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1answer
88 views

Proving that $\sf Add$$(\aleph_\omega , 1)$ collapses cardinals $\leq \aleph_\omega$

First, let me fix some notation. $\sf Fn$$(I, J, \kappa) = $ the poset of all partial functions $p$ such that $|p| < \kappa$, dom$(p) \subseteq I$ and rng$(p) \subseteq J$. $\sf Add$$(\kappa, ...
5
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1answer
88 views

$\kappa\psi (x,X)\leq \psi (x,X)$

The $\kappa$-pseudocharacter $\kappa\psi (x,X)$ of a space $X$ at a point $x\in X$ is the smallest infinite cardinal number $\tau$ such that there exists a family $\gamma$ of $\kappa$-sets in $X$ ...
5
votes
1answer
119 views

What indexes do the subgroups of $\mathrm{GL}_n(\Bbb C)$ have?

Let $B_n\subset\mathrm{GL}_n(\Bbb C)$ be the group of invertible upper-triangular matrices. What is the index $[\mathrm{GL}_n(\Bbb C):B_n]?$ (By index I mean the cardinality of a coset space.) In ...
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1answer
560 views

How can one rigorously determine the cardinality of an infinite dimensional vector space?

Suppose $V$ is a vector space over a scalar field $F$. If $\dim(V)=n$, then $|V|=|F|^n$. How can I rigorously determine the cardinality of $V$ when $V$ is infinite dimensional? My thought was that if ...
5
votes
1answer
119 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 $\kappa$?
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1answer
299 views

Cardinality of sets of functions with well-ordered domain and codomain

I would like to determine the cardinality of the sets specified bellow. Nevertheless, I don't know how to approach or how to start such a proof. Any help will be appreciated. If $X$ and $Y$ are ...
5
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1answer
168 views

Does any uncountable set contain two disjoint uncountable sets?

Is it true that for any uncountable $S$, there exits two uncountable subsets $S_1,S_2 \subseteq S$ with $S_1 \cap S_2 = \emptyset$? I can find no counter example, but no proof either. I am aware of ...
5
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1answer
110 views

Partial order on cardinalities without the axiom of choice

Cardinality can still be defined without choice, e.g. as equivalence class of equipotent sets, see Defining cardinality in the absence of choice. Injections define partial order on cardinalities by ...
5
votes
1answer
80 views

How to show that $\mathfrak s \leq \mathfrak d$

I am trying to understand why $\mathfrak s \leq \mathfrak d$. Can anyone state a proof of it? I have a proof , which I don't understand yet. My question regarding that proof is here below: At the ...
5
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1answer
81 views

A verification for a proof that $\omega_1 \leq \mathfrak s$

I am trying to prove that $\omega_1 \leq \mathfrak s$ where $\mathfrak s$ is the splitting number which is the smallest cardinality of any splitting family. This statement was left as an ...
5
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1answer
160 views

Jech's proof of Silver's Theorem on SCH

Jech's textbook proves Silver's Theorem–that if the Singular Cardinals Hypothesis holds for all singular cardinals of countable cofinality, then it holds for all singular cardinals–by breaking up the ...
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1answer
121 views

Differences in worlds with and without $\aleph_0<|S|<2^{\aleph_0}$

Paul Cohen told us that whether or not there is $S$ with \begin{equation} \aleph_0<|S|<2^{\aleph_0} \end{equation} cannot be decided within ZFC, and hence it is reasonable to work in two ...
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1answer
162 views

Orders of subgroups of Infinite Profinite Groups

This question is in some sense equivalent to my question here. A proof would answer that question in the case when the base field is perfect. Let $G$ be a profinite group of cardinality $\kappa$, ...