# Questions tagged [cardinals]

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.

2,429 questions
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### How do we know an $\aleph_1$ exists at all?

I have two questions, actually. The first is as the title says: how do we know there exists an infinite cardinal such that there exists no other cardinals between it and $\aleph_0$? (We would have ...
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### Why is $\omega$ the smallest $\infty$?

I am comfortable with the different sizes of infinities and Cantor's "diagonal argument" to prove that the set of all subsets of an infinite set has cardinality strictly greater than the set itself. ...
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### Doesn't the unprovability of the continuum hypothesis prove the continuum hypothesis? [duplicate]

The Continuum Hypothesis say that there is no set with cardinality between that of the reals and the natural numbers. Apparently, the Continuum Hypothesis can't be proved or disproved using the ...
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### Why is the Continuum Hypothesis (not) true?

I'm making my way through Thomas W Hungerfords's seminal text "Abstract Algebra 2nd Edition w/ Sets, Logics and Categories" where he makes the statement that the Continuum Hypothesis (There does not ...
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### Refuting the Anti-Cantor Cranks

I occasionally have the opportunity to argue with anti-Cantor cranks, people who for some reason or the other attack the validity of Cantor's diagonalization proof of the uncountability of the real ...
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### Cardinality of set of real continuous functions

The set of all $\mathbb{R\to R}$ continuous functions is $\mathfrak c$. How to show that? Is there any bijection between $\mathbb R^n$ and the set of continuous functions?
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### Overview of basic results on cardinal arithmetic

Are there some good overviews of basic formulas about addition, multiplication and exponentiation of cardinals (preferably available online)?
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### Is symmetric group on natural numbers countable?

I guess it is too difficult a question to ask about the cardinality of $S_{\mathbb{N}}$ so I would like to ask whether it is countable or not. I tried to prove it is uncountable somewhat mimicking ...
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### Cardinality of the set of all real functions of real variable

How does one compute the cardinality of the set of functions $f:\mathbb{R} \to \mathbb{R}$ (not necessarily continuous)?
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### For every infinite $S$, $|S|=|S\times S|$ implies the Axiom of choice

How to prove the following conclusion: [For any infinite set $S$,there exists a bijection $f:S\to S \times S$] implies the Axiom of choice. Can you give a proof without the theory of ordinal numbers....
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### Is cardinality a well defined function?

I was wondering if the cardinality of a set is a well defined function, more specifically, does it have a well defined domain and range? One would say you could assign a number to every finite set, ...
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### Defining cardinality in the absence of choice

Under ZFC we can define cardinality $|A|$ for any set $A$ as $$|A|=\min\{\alpha\in \operatorname{Ord}: \exists\text{ bijection } A \to \alpha\}.$$ This is because the axiom of choice allows any ...
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### Proof that the number of proofs is countably infinite

I think that the number of mathematical statements is countably infinite, since each statement is a finite string of finitely many symbols, but i am not sure how to prove it. Once i prove that, i can ...
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### Cardinality of all cardinalities

Let $C = \{0, 1, 2, \ldots, \aleph_0, \aleph_1, \aleph_2, \ldots\}$. What is $\left|C\right|$? Or is it even well-defined?
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### Is there a set whose power set is countably infinite? [duplicate]

Does there exist a set whose power set is countably infinite? I know for sure that if a set has a finite number of elements, then its power set must have a finite number of elements, and if a set has ...
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### How to show $(a^b)^c=a^{bc}$ for arbitrary cardinal numbers?

One of the basic (and frequently used) properties of cardinal exponentiation is that $(a^b)^c=a^{bc}$. What is the proof of this fact? As Arturo pointed out in his comment, in computer science this ...
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### Are there number systems corresponding to higher cardinalities than the real numbers?

As most of you know, the set $\omega$ with cardinality $\aleph_0$ corresponds to what we normally know as the natural numbers $\mathbb{N}$, and the set $\mathcal{P}(\omega)$ with cardinality $\aleph_1$...
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### How to formulate continuum hypothesis without the axiom of choice?

Please correct me if I'm wrong but here is what I understand from the theory of cardinal numbers : 1) The definition of $\aleph_1$ makes sense even without choice as $\aleph_1$ is an ordinal number (...
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### The cardinality of a countable union of countable sets, without the axiom of choice

One of my homework questions was to prove, from the axioms of ZF only, that a countable union of countable sets does not have cardinality $\aleph_2$. My solution shows that it does not have ...
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### For any two sets $A,B$ , $|A|\leq|B|$ or $|B|\leq|A|$

Let $A,B$ be any two sets. I really think that the statement $|A|\leq|B|$ or $|B|\leq|A|$ is true. Formally: $$\forall A\forall B[\,|A|\leq|B| \lor\ |B|\leq|A|\,]$$ If this statement is true, what ...
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### Does $\mathfrak{(p<q)\land(r<s)}$ imply $\mathfrak{p^r<q^s}$?

Does $\mathfrak{(p<q)\land(r<s)}$ imply $\mathfrak{p^r<q^s}$, where $\mathfrak{ p,q,r,s}$ are cardinal numbers? Is it possible to prove in $\mathsf{ZFC}$ that there is a counterexample?
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### Covering $\mathbb R^2$ with function graphs

Suppose we have a countable family of function graphs (each function is $\mathbb R\to\mathbb R$, not necessary continuous). Obviously, they cannot cover the whole plane $\mathbb R^2$, because they ...
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### Why study cardinals, ordinals and the like?

Why is the study of infinite cardinals, ordinals and the like so prevalent in set theory and logic? What's so interesting about infinite cardinals beyond $\aleph _0$ and $\mathfrak{c}$? It seems ...
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### 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 ...
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### Do you need the Axiom of Choice to accept Cantor's Diagonal Proof?

Math people: It is my understanding that set theorists/logicians compare cardinalities of sets using injections rather than surjections. Wikipedia defines countable sets in terms of injections. ...
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### What do the cosets of $\mathbb{R} / \mathbb{Q}$ look like?

$\newcommand{\R}{\Bbb R}\newcommand{\Q}{\Bbb Q}$ Looking at the group of real numbers under addition $(\R, +)$ it contains the (normal) subgroup of rational numbers $(\Q, +)$. I am wondering how to ...
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### How many non-differentiable functions exist?

The size of the set of functions that map $\mathbb{R}\to \mathbb{R}$ equals $(\#\mathbb{R})^{\#\mathbb{R}}$. How many non-differentiable functions are there in this set?
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### Why hasn't GCH become a standard axiom of ZFC?

I've never seen a text that includes GCH in the ZFC axioms. I presume this means that GCH has not achieved widespread acceptance. This seems surprising to me, given that: The cardinal numbers ...
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### Do the real numbers and the complex numbers have the same cardinality?

So it's easy to show that the rationals and the integers have the same size, using everyone's favorite spiral-around-the-grid. Can the approach be extended to say that the set of complex numbers has ...
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### Is aleph-$0$ a natural number?

Would I be right in saying that $\aleph_0 \in \mathbb N$? Or would it be a wrong thing to do?