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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126 views

What is the cardinality of $\Bbb{R}^L$?

By $\Bbb{R}^L$, I mean the set that is interpreted as $\Bbb{R}$ in $L$, Godel's constructible universe. For concreteness, and to avoid definitional questions about $\Bbb{R}$, I'm looking at the set ...
6
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3answers
652 views

Easiest way to prove that $2^{\aleph_0} = c$

$\aleph_0$ is the cardinality of the set of natural numbers, $\aleph_0 = |N|$. $c$ is the cardinality of the continuum, i.e. the set of real numbers $c = |R|$. I know that $|P(A)| = 2^{|A|}$. This ...
6
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2answers
156 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!
6
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3answers
720 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 ...
6
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3answers
195 views

Prove: $B=\{x\in \Bbb R\mid x^2\in\Bbb Q\}$ is countable

Prove: The set $B=\{x\in \Bbb R\mid x^2\in\Bbb Q\}$ is countable I have this idea but I'm not sure if it's correct: We know $\Bbb Q$ is countable so we can list $\Bbb Q$ as $\Bbb Q= \{ ...
6
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3answers
139 views

What questions become answerable/computable given an uncountable character set?

Having reached the concluding portion of my first course in real analysis, one subject that I feel was not adequately addressed was the issue of cardinalities. This is a subject I was interested in ...
6
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2answers
280 views

Confusion about cofinality

I'm confused about the notion of the cofinality of a cardinal. Since I think the source of the confusion is the Von Neumann cardinal assignment, my first question is: Question 0. Is there an article ...
6
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2answers
214 views

Is it viable to ask in an infinite set about the Cardinality?

Can you ask given an infinite set about its cardinality? Does an infinite set have a cardinality? So, for example, what would be the cardinality of $+\infty$?
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2answers
404 views

Fodor's lemma on singular cardinals

Fodor's lemma asserts that if $\kappa$ is a regular and uncountable cardinal, then if $f(\alpha)<\alpha$ for a stationary subset of $\kappa$, then it is constant on stationary subset. Suppose ...
6
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1answer
1k views

Cardinality of a set that consists of all existing cardinalities

What is the easiest way to prove (if possible, without using ordinals etc. as my current math understanding of set theory counts only cardinals, and countable & uncountable sets) that the number ...
6
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2answers
84 views

Why was $\aleph$ (aleph) chosen for infinities?

Why did Cantor choose a letter from the Hebrew alphabet to represent infinities, rather than using some Greek letter?
6
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2answers
162 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 ...
6
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2answers
146 views

Skolem's paradox and models in set theory

Say you have a model $M$ (of $\mathsf{ZF}$) containing a set $S$ that you know is countable outside the model but the enumeration is missing from the model so that $S$ appears to be uncountable in ...
6
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3answers
2k views

Cardinality of a set A is strictly less than the cardinality of the power set of A

I am trying to prove the following statement but have trouble comprehending/going forward with some parts! Here is the statement: If $A$ is any set, then $|A|$ $<$ $|P(A)|$ Here is what I ...
6
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3answers
2k 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?
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2answers
163 views

Sum of cardinals without AC

Let $A$ and $B$ be infinite sets. To show $|A\cup B|=\max\{|A|,|B|\}$ we need AC. Now let us assume $|A|<|B|$. Can we show $|A\cup B|=|B|$ without AC?
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3answers
173 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 ...
6
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2answers
185 views

In ZF, how would the structure of the cardinal numbers change by adopting this definition of cardinality?

In ZFC, a good way of ordering sets by cardinality is by leveraging the notion of an injection. We define: $$X \lesssim Y \leftrightarrow \mbox{ there exists an injection } X \rightarrow Y.$$ ...
6
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1answer
226 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?
6
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1answer
414 views

Is there a way to define the “size” of an infinite set that takes into account “intuitive” differences between sets?

The usual way to define the "size" of an infinite set is through cardinality, so that e.g. the sets $\{1, 2, 3, 4, \ldots\}$ and $\{0, 1, 2, 3, 4, \ldots\}$ have the same cardinality. However, is this ...
6
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2answers
204 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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2answers
357 views

mahlo and hyper-inaccessible cardinals

Wikipedia states that a Mahlo cardinal is hyper-inaccessible, hyper-hyper-inaccessible, etc. Is this a characterisation of Mahlo? If not what about "alpha = hyper^alpha-inaccessible" with the obvious ...
6
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1answer
103 views

Well-foundedness of cardinals and the axiom of choice

Without axiom of choice it is not generally true that the class of all cardinal (in this question we consider Scott cardinal rather than cardinals as ordinals) is not well-founded under the ordinary ...
6
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2answers
130 views

Does $\sf GCH$ imply that every uncountable cardinal is of the form $2^\kappa$?

I think that this is a popular fallacy that GCH implies that every uncountable cardinal is of the form $2^\kappa$ for some $\kappa$. I think it does imply that for successor cardinals only. It cannot ...
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2answers
979 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 ...
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2answers
212 views

What happens after the cardinality $\mathfrak{c}$?

While having measure theory this year the following came in my mind: When we go from finite objects to infinite we "lose" a lot of properties. For example the summation isn't well defined ...
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3answers
1k views

Number of countable subsets of $\mathbb{R}$

More generally, if a set $S$ has cardinality $\mathfrak{m}$, how many of its subsets have cardinality $\mathfrak{n}$? Clearly there are at least $2^\mathfrak{n}$ such subsets. I don't see how many ...
6
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1answer
119 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, ...
6
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1answer
103 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 ...
6
votes
1answer
87 views

How to show $\kappa^{cf(\kappa)}>\kappa$?

For $\kappa \geq \omega$, which is a cardinal, then how to show $\kappa^{cf(\kappa)}>\kappa$? My idea: When $\kappa=\aleph_\omega$, then $cf(\kappa)=\omega$, is $\kappa^\omega>\kappa$? It seems ...
6
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2answers
2k views

How do you prove the trichotomy law for cardinal numbers?

Law of trichotomy is that for any two cardinals $a$ and $b$, exactly one of the conditions $a<b$, $a=b$, or $a>b$ holds.
6
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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 $\kappa$?
6
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1answer
197 views

How to prove that a regular cardinal cannot be expressed as a union of sets with less cardinality?

My question is about the following: Using the Axiom of Choice show that: If $\kappa\ge\omega$ is a regular cardinal, $\gamma\le\kappa$, and $\langle A_\alpha\mid\alpha\lt\gamma\rangle$ is a ...
6
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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 ...
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2answers
699 views

Let $X$ and $Y$ be countable sets. Then $X\cup Y$ is countable

Since $X$ and $Y$ are countable, we have two bijections: (1) $f: \mathbb{N} \rightarrow X$ ; (2) $g: \mathbb{N} \rightarrow Y$. So to prove that $X\cup Y$ is countable, I figure I need to define ...
6
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1answer
416 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 ...
6
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1answer
485 views

On cardinality of equivalence classes

Say $R$ is the set of all Riemann integrable functions $f:[a,b]\rightarrow \mathbb{R}$. We define an equivalence relation in $R$ as follows: $f$ and $g$ are said to be integrally equivalent iff they ...
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2answers
153 views

Effective cardinality

Consider $X,Y \subseteq \mathbb{N}$. We say that $X \equiv Y$ iff there exists a bijection between $X$ and $Y$. We say that $X \equiv_c Y$ iff there exist a bijective computable function between ...
6
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1answer
66 views

What is the cardinality of an element of an free ultrafilter?

Let $U$ be a free ultrafilter on a set $X$. I want to prove that the cardinality of every element $u\in U$ is equipotent to $X$. Is that true? Or does it lack some hypothesis?
6
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1answer
695 views

List of explicit enumerations of rational numbers [closed]

A well-known mathematical fact is that the rational numbers are countable, i.e. there is a bijective function $$f:\mathbb{N}\rightarrow \mathbb{Q}$$ I am interesting in making a list of all explicit ...
6
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1answer
145 views

$\kappa <\operatorname{cf}(2^\kappa)$ without König's inequality

How can I prove $\kappa<\operatorname{cf}(2^\kappa)$ inequality without using König's inequality? We got this as a practice exercise, but I don't know how to approach this without König. Any hint ...
6
votes
1answer
216 views

How to explain that $\Bbb{R}$ is not countable to a non-mathematician

What is the best way to explain that $\Bbb{R}$ is not countable assuming that the audience is formed of people who are not mathematicians? I ask this because these days I'm in a debate with someone ...
6
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1answer
214 views

Exercise on partially ordered sets from Kunen's *Set Theory*

This problem is Exercise (F4) from Chapter VII of Kunen's text. Apparently, it is a result of Tarski. It goes as follows: Let $ \mathbb{P} $ be a partial order. Define $ \text{c.c.}(\mathbb{P}) $ ...
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1answer
157 views

Dominating strategically $\omega_1$ reals

For a given $\kappa > \omega$, define the game $d(\kappa)$ that runs for $\omega$ stages as follows: At stage $n$, player I chooses a sequence of elements of $\omega$, $g_n$ of length $\kappa$, and ...
6
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1answer
408 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 ...
6
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1answer
419 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 ...
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0answers
138 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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3answers
488 views

Are there fields and ordered fields of every infinite cardinality?

On the top of my head, I cannot think of any fields of cardinality more than that of the reals. (It is known that the process of algebraic closure does not increase the cardinality of an infinite ...
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3answers
1k 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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2answers
769 views

Why are infinite cardinals limit ordinals?

My book states this as obvious, but it isn't so trivial to me. thanks