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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7
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1answer
151 views

What would a world where $\mathsf{CH}$ is false look like?

My question is a little more specific than the title may lead to believe. In the article The set-theoretic multiverse (J.D. Hamkins), the author writes the following: [...] the continuum is ...
7
votes
1answer
216 views

Question about the order of a Stationary subset of $ \kappa$

Greets I'm trying to prove one part of exercise 8.14 of Jech's "Set Theory", namely that if $o(k)\geq k$, then $k$ is weakly inaccessible, where $\kappa$ is regular; $o(\kappa)$ is defined as ...
7
votes
1answer
322 views

powers of singular cardinals

I am trying to solve the following two problems: 1) if $\beta <\omega_1$, $2^{\aleph_1}<\aleph_{\omega_1}$, and $\aleph_\alpha^{\aleph_0} \leq \aleph_{\alpha +\beta}$ for a stationary set of ...
7
votes
2answers
867 views

Non-existence of a surjection $\aleph_n \to \aleph_{n+1}$, without the axiom of choice

Firstly, let's establish what exactly I mean by these symbols. Let $\omega_0 = \{ 0, 1, 2, \ldots \}$, where $0, 1, 2, \ldots$ are the usual von Neumann representations of the natural numbers. Let $n$ ...
7
votes
1answer
104 views

For an infinite cardinal $\kappa$, $\aleph_0 \leq 2^{2^\kappa}$

I'm trying to do a past paper question which states: $$ \text{For all infinite cardinals $\kappa$, we have } \aleph_0 \leq 2^{2^\kappa}. $$ I'm supposed to be able to do this without the axiom of ...
7
votes
1answer
152 views

weak consequence of GCH

Can ZFC prove that there is a regular uncountable cardinal $\kappa$ such that $2^{<\kappa} < 2^\kappa$? Note, if the answer is no, it would require a strong global violation of SCH, so large ...
7
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1answer
91 views

Characterizing categories by size

Usually one distinguishes five classes of categories by size, and there are examples for all of them: finite categories locally finite categories small categories locally small categories large ...
7
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1answer
139 views

Why should the union of such a family of sets have cardinality $\aleph_1?$

Let $\mathcal A$ be a family of infinite countable sets which is linearly ordered by inclusion and such that $\bigcup\mathcal A$ is uncountable. I have to prove that $|\bigcup\mathcal A|=\aleph_1.$ I ...
7
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1answer
767 views

Examples of sets whose cardinalities are $\aleph_{n}$, or any large cardinal. (not assuming GCH)

One of the answers to this question indicates that large cardinals are useful for destructive testing of set theory. That aside, and not assuming GCH, are there any sets known that have a cardinality ...
7
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1answer
152 views

Constructing a Banach space of cardinality $\beth_{\omega+1}$

This is related to yesterday's question Constructing a vector space of dimension $\beth_\omega$; it's the next exercise (I.13.35 (a)) in Kunen's Set Theory. Let $B_0 = \ell^1$ and let $B_{n+1} = ...
7
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1answer
97 views

$V=L[A]$ implies GCH for $A\subset \aleph_1$

On page 14 of the introduction to Vol. II of Gödel's collected works: By a slightly more difficult argument one can show that GCH continues to hold if $V=L[a]$ and $a\subseteq\aleph_1$. Does ...
7
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1answer
142 views

$(\beth_{\omega})^\omega=\beth_{\omega+1}$

I'm trying to show that $(\beth_{\omega})^\omega=2^{\beth_\omega}$. This is an exercise in Kunen where he suggests to encode subsets of $\beth_\omega$ with functions from ...
7
votes
1answer
135 views

Instance of Continuum Hypothesis implying cardinal inequality

I'm currently trying to solve Exercise 5.27 of Jech's Set Theory (3rd Millennium ed.), viz: If $2^{\aleph_1}=\aleph_2$, then $\aleph_{\omega}^{\aleph_0} \ne \aleph_{\omega_1}$. The presumption ...
7
votes
0answers
104 views

Raising a partial function to the power of an ordinal

Consider a set $X$, and let $f : X \rightarrow X$ denote a partial function. Then for natural $n$, we can define $f^n$ as iterated composition, e.g. $f^2 = f \circ f$. Now suppose that $X$ is also ...
7
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0answers
116 views

AC iff $P(\delta)$ can be well-ordered for all $\delta\in{\bf On}$ [duplicate]

I'm trying to do the following exercise: Exercise. Assume ZF. Then AC is equivalent to the statement that $P(\delta)$ can be well-ordered for all $\delta\in{\bf On}$. I'm struggling with the ...
7
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0answers
239 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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5answers
2k views

Are there fewer reals on $(0, 1)$ than on $(1,\infty)$?

I know that the cardinality of the sets of real numbers $(0, 1)$ and $(1, \infty)$ are equal. So what is the fallacy in this argument? For every real on $(0, 1)$, we can add any integer $n$ to it ...
6
votes
3answers
847 views

How do we prove the existence of uncountably many transcendental numbers?

I know how to prove the countability of sets using equivalence relations to other sets, but I'm not sure how to go about proving the uncountability of the transcendental numbers (i.e., numbers that ...
6
votes
5answers
388 views

cardinality of the set of $ \varphi: \mathbb N \to \mathbb N$ such that $\varphi$ is an increasing sequence

I know that the set of functions $ f:\mathbb N \to \mathbb N$ is uncountable, but what if we consider only $f$ such that $f$ is increasing? I want to know if this set is countable D: and also the case ...
6
votes
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 $\mathbb{R}$. ...
6
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4answers
2k views

What is the set of all functions from $\{0, 1\}$ to $\mathbb{N}$ equinumerous to?

What is the set of all functions from $\{0, 1\}$ to $\mathbb{N}$ equinumerous to? I have figured out the question when it's the other way around, but I am not making any progress here. The worst thing ...
6
votes
3answers
348 views

What is the first cardinal number which is grearter than continuum?

What is the first cardinal number which is grearter than continuum? We denote it by ? Thanks very much.
6
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3answers
603 views

Can we distinguish $\aleph_0$ from $\aleph_1$ in Nature?

Can we even find examples of infinity in nature?
6
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3answers
740 views

How to prove cardinality equality ($\mathfrak c^\mathfrak c=2^\mathfrak c$)

How do I prove this cardinality equality:$\mathfrak c^\mathfrak c=2^\mathfrak c$ I have failed to prove this after lots of trail - but I am certain it's true How can I prove this?
6
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2answers
366 views

Proving existence of a surjection $2^{\aleph_0} \to \aleph_1$ without AC

I'm quite sure I'm missing something obvious, but I can't seem to work out the following problem (web search indicates that it has a solution, but I didn't manage to locate one -- hence the ...
6
votes
2answers
369 views

prove cardinality rule $|A-B|=|B-A|\rightarrow|A|=|B|$

I need to prove this $|A-B|=|B-A|\rightarrow|A|=|B|$ I managed to come up with this: let $f:A-B\to B-A$ while $f$ is bijective. then define $g\colon A\to B$ as follows: $$g(x)=\begin{cases} ...
6
votes
2answers
549 views

What is the Cardinality of the Nameable Numbers?

Having just finished "Meta Math!" (Chaitin), I came across an interesting observation on infinite sets that I hadn't seen before. If I'm correct (and please let me know if I'm not): 1] There are ...
6
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3answers
748 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?
6
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2answers
735 views

Cardinality of sets of subsets of $\mathbb{N}$

If we dont assume CH, is there a procedure to construct or define a set of subsets of $\mathbb{N}$ such that we cannot prove it to be of cardinality $\aleph_0$ or $\aleph_1$? Or if we assume not CH, ...
6
votes
4answers
104 views

Is the logarithm of $\aleph_0$ infinite?

In classical mathematics $2^{\aleph_0}=\aleph_1$, right? So if $2^x=\aleph_0$, what does $x$ equal? In other words, can we define a logarithm for $\aleph_0$, and what should it be. Is it infinite? ...
6
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1answer
249 views

is there a cardinality between the rational and the irrationals?

Was asked this question, and I have no idea about how to start proving it. Could someone give me some good reference material to start with.
6
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5answers
344 views

Are there more real numbers than we can actually imagine?

I mean, if we could imagine all the real numbers then we could assign each number a finite sentence (or a finite book). Since the set of the finite books is countable then the set of real numbers ...
6
votes
6answers
1k views

Proof that aleph null is the smallest transfinite number?

The wikipedia page on the cardinal numbers says that $\aleph_0$, the cardinality of the set of natural numbers, is the smallest transfinite number. It doesn't provide a proof. Similarly, this page ...
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3answers
569 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
votes
2answers
48 views

Let $X \subseteq \mathbb R$ and $X$ has same cardinality as $\mathbb R$ , does there always exist a continuous surjection from $\mathbb R$ onto $X$ ?

Let $X \subseteq \mathbb R$ and $X$ has same cardinality as $\mathbb R$ , does there always exist a continuous surjection from $\mathbb R$ onto $X$ ? ( I know that there need not always be a ...
6
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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!
6
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3answers
622 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
votes
1answer
119 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
votes
2answers
256 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
204 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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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?
6
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2answers
3k views

Why do the rationals, integers and naturals all have the same cardinality?

So I answered this question: Are all infinities equal? I believe my answer is correct, however one thing I couldn't explain fully, and which is bugging me, is why the rationals $\mathbb Q$, integers ...
6
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2answers
379 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
942 views

Cardinality of a set that consists of all existing cardinalities

I have taken a look at the following topics: number of infinite sets with different cardinalities Cardinality of all cardinalities Are there uncountably infinite orders of infinity? Types of ...
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2answers
139 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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2answers
156 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?
6
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3answers
167 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
votes
2answers
171 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
votes
1answer
208 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?
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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 ...