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.

Cardinality is a notion of size for sets, usually denoted by $|A|$ as the "cardinality of $A$". With finite sets the cardinality is simply the number of elements which are members of a set.

Dealing with infinite sets we can measure them in different ways. Cardinal numbers are very natural in the sense that they do not require extra structure (such as relations and operations defined on the set to be preserved).

In formal terms, suppose $f\colon A\to B$ (i.e. $f$ is a function whose domain is $A$ and its range is a subset of $B$).

We say that $f$ is injective if $f(a)=f(b)$ implies $a=b$; we say $f$ is surjective if its range is all $B$, namely for any $b\in B$ there is $a\in A$ such that $f(a)=b$.

If $f$ is both surjective and injective we say that $f$ is a bijection from $A$ to $B$. The inverse of a bijection is also a bijection.

Now we can define an equivalence relation on sets, $A\sim B$ if and only if there is some $f\colon A\to B$ which is a bijection.

Assuming the Axiom of Choice, we have that every set can be well ordered, and therefore there is a least ordinal which is equivalent to $A$, so we can assign it as a canonical representative for the equivalence class, usually denoted by $\aleph_\alpha$ where $\alpha$ is an ordinal, or as general Greek letters such as $\kappa,\lambda$.

Before defining the $\aleph$ numbers we need to define initial ordinals. Let $\alpha$ be an ordinal, if there is no $\beta<\alpha$ and $f\colon\alpha\to\beta$ which is a bijection, then $\alpha$ is called an initial ordinal.

The $\aleph$ numbers are defined by transfinite induction as:

  1. $\aleph_0 = |\omega| = \omega$ (note that $\omega$ is an initial ordinal),
  2. $\aleph_{\alpha+1} = \aleph_\alpha^+$ (where the $\cdot^+$ means the smallest initial ordinal above the one defined for $\aleph_\alpha$)
  3. If $\beta$ is a limit ordinal, then $\displaystyle\aleph_\beta = \bigcup_{\delta<\beta}\aleph_\delta$ (It is easy to verify that the union of initial ordinals is an initial ordinal).

The confinality of an $\aleph$ number is the minimal cardinality of a set which is unbounded in the initial ordinal matching the $\aleph$ number.

A cardinal is called regular if its cofinality is itself, otherwise it is called singular.

Example: $\aleph_0$ is regular, because for a set to be unbounded below $\omega$ it cannot be finite.

$\aleph_1$ is also regular, every ordinal below $\omega_1$ is countable, and the union of countably many countable ordinals is just countable - which is still below $\aleph_1$.

Example: $\aleph_\omega$ is singular, recall $\displaystyle\aleph_\omega=\bigcup_{n<\omega}\aleph_n$. Therefore the set $\{\omega_n\mid n<\omega\}$ (the collection of initial ordinals whose cardinality is less than $\aleph_\omega$) is unbounded, and its cardinality is merely countable.

The question whether or not there exists $\aleph_\delta$ such that $\delta$ is a limit ordinal, but $\aleph_\delta$ is regular is unprovable in ZFC. It is known that it is consistent that there are none, but unknown that it is inconsistent that there are. Cardinals with this property are called Large cardinals and are used for consistency proofs.


In the absence of choice we can no longer have canonical representatives for the equivalence classes, and things become tricky. The class of cardinals can still be defined, however in a slightly different way - usually Scott's trick.

However, to show how things can break down it is consistent with ZF that there is no choice function on the equivalence classes (i.e. you cannot have canonical representatives).

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Few questions about the basics of Cardinality

I am looking for some help to either conform that my reasoning is sound, or to please elaborate to me more on the subject so I can gain a better understanding. I am studying some from my class notes, and some from other sources. Suppose we are able…
Quality
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Cardinality of $X^n$

I asked the following question before: Is this proof by induction that $|\Bbb Q^n|=\aleph_0$ correct? I want to know if the proccess I did that can be generalized to the case $|X|=\kappa; \kappa$ being any infinite cardinal. This would be as…
YoTengoUnLCD
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Can it be proved without the axiom of choice that every cardinal is comparable with every finite cardinal?

Can it be proven in ZF, without using the axiom of choice, that every finite set is a universal size comparator, meaning, is comparable with every set in terms of size? And what is the proof?
user107952
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Cardinal Exponentiation Inequality

Let $\lambda, \kappa$ be infinite cardinals with $\lambda<\kappa$, what is known about $\kappa^\lambda$? specially in the case either $\kappa$ is regular. Or is there very little that can be answered about this question under ZFC (Again I'm mostly…
user185596
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What is the cardinality of the set of roots of unity?

Consider the geometric interpretation of "roots of unity": My intuition says that you can place arbitrarily many equidistant points on the unit circle and catch every point that lies on it. Therefore every $z \in \mathbb{C}$ that lies on the unit…
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a question about analysis, how to find the largest cardinality in the following examples

This is a GRE math question: My thoughts: I guess as for the cardinality, (A)=(B) and (D)=(E),but I couldn't prove whether it is true or not. Also, how to find the cardinality of (C), can someone tell me how to analyze this problem? I still have no…
python3
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The size of the set of continuous function of period T

I have a naive question. The Fourier series give an injection between continuous function of period $T$ and the set of real valued sequences. But, don't we expect the set of continuous function of periode $T$ to be much bigger than the set of…
Chevallier
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Cardinality of the power set of the set of all primes

Please show me what I am doing wrong... Given the set $P$ of all primes I can construct the set $Q$ being the power set of P. Now let $q$ be an element in $Q$. ($q = \{p_1,p_2,p_3,\ldots\}$ where every $p_n$ is an element in $P$.) Now I can map…
iolo
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Cardinality of $A^B$ when $A > B \ge \aleph_0$

For an infinite cardinal A, then if $B$ is finite $A^B = A$ If $B$ is infinite and $B \ge A$ then $A^B \ge A^A \ge 2^A > A$ What if $B$ is infinite, but $B < A$, i.e. $A > B \ge \aleph_0$. Is there a relationship between $A^B$ and $A$ in this case…
Tom Collinge
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Confused about one to one functions and cardinality

There is something that I'm not getting about functions and cardinality of sets. I've read the following: If $F$ is a one to one function, then $F^{-1}$ (the inverse) is also a one-to-one function. Because when I read about cardinality of sets,…
J. Bend
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Is the set of all of possible uncountable cardinalities uncountably or countably infinite?

I assume that the number of unique possible cardinalities that are uncountably infinite is either uncountable or countable because it is possible to take the powerset of each set, resulting in an uncountable number of elements; thus, there would be…
will
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Cardinality of $P_\mathfrak{c}(\mathbb R)$

Calculate the cardinality of $\mathcal P_\mathfrak{c}(\mathbb R)=\{\mathcal R \in \mathcal P(\mathbb R) \, /\, \#(\mathcal R)=\mathfrak{c}\}$. I thought that instead of $\mathcal P_\mathfrak{c}(\mathbb R)$ I should think in $\mathcal P_c([0,1))$ so…
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If $n < \aleph^*(m)$, then $n < 2^m$.

Without $AC$ Let $\aleph^*(m)$ be the least aleph that $\not\leq^* m$. I need a help or hint that if $n < \aleph^*(m)$, then $n < 2^m$. $a \leq^* b$ means we can define a surjective map from $b$ onto $a$.
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Cardinal exponentation

Let $\lambda, \kappa$ be infinite cardinals. Does it follow from the inequality $\kappa\lt2^\lambda$ that $2^\kappa\lt2^{2^\lambda}$?
denhaag
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Cardinality of a club

I proved that a club $C$ in $\kappa$ has the same cardinality as $\kappa$. Is it really true ? Thanks.
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