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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A theorem about cardinal numbers(an inequality)

Theorem. Let $\{ A_k | k \in K \}$ be a collection of sets indexed by the $K$, with $|K| = \kappa$. If $\forall k \in K \ \ |A_k| \leq \lambda$, then $|\bigcup\limits_{k \in K} A_k| \leq ...
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1answer
36 views

Independent families of sets

I'm having a difficulty understanding some exercises related to independent families of sets. Recall that $ \mathcal{A} $ is $\lambda$-independent if for any disjoint $ P, Q \in \mathcal{A} : |P|, |Q| ...
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1answer
19 views

Generalized version of uncountable minus countable is uncountable

I think my question is generalized version of Uncountable minus countable set is uncountable I have to show: if $A$ is an infinite set, and $B$ is a subset of $A$, which satisfies $|B|<|A|$, then ...
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1answer
53 views

The generalized continuum hypothesis can't fail first at $\omega_{\omega_{1}}$

I am willing to prove that the GCH cannot first fail at a singular cardinal. For this purpouse I am following the strategy outlined by Kunen in his 2013 book (see Exercises III.6.16-6.17-6.18-6.19). I ...
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2answers
37 views

Size of cardinal without choice

How can we show that $ \aleph_0 \leq 2^{2^\kappa}$ for any infinite cardinal $\kappa$ without using the Axiom of Choice? By Cantor's Theorem we can easily show that if $ \aleph_0 > 2^{2^\kappa}$, ...
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1answer
20 views

Associativity of cardinal sum

I'm stuck with the first exercise of chapter 9 from Jech and Hrbacek Introduction to set theory. It states: If $J_i\,(i\in I)$ are mutually disjoint sets and $J=\bigcup_{i\in I}J_i$, and if ...
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3answers
60 views

What is cardinality of set of all intervals (a,b), where a, b are rational numbers?

We have set $S=\{ (a,b) | a,b \in \mathbb{Q}\}$ And we know that $(a,b)\sim \mathbb{R}$ , so $k((a,b))=c$. And $\mathbb Q \sim \mathbb N$, so $k(\mathbb Q)=\aleph_0$. I don't know how to put all ...
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1answer
12 views

Increasing sequence of cardinals and cardinal $a^{\aleph_0}$

I've got the following problem: let $m_0<m_1<m_2<\cdots$ be an increasing sequence of cardinals. Prove that the sum $m_0+m_1+m_2+\cdots$ diffiers from $a^{\aleph_0}$ for any cardinal $a$. ...
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1answer
10 views

Intersection of decreasing family of $\kappa-$many sets

Let's consider $\kappa$ an uncountable regular cardinal and $\lambda<\kappa$. Given any decreasing family $\{A_\alpha\}_{\alpha<\lambda}$ of sets with $\sharp A_\alpha=\kappa$, does it true that ...
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1answer
32 views

Proving cardinality of coproduct presentation is unique without choice?

The definition of an extensive category immediately implies that given two coproduct decompositions indexed by sets of equal cardinality, if the coproduct objects are isomorphic compatibly with their ...
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Cardinality of Galois groups

We know that no Galois group of a Galois extension is countable. The question is: which cardinalities are possible for a Galois group? (i.e. for profinite groups?) I suspect that the theory of Galois ...
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1answer
37 views

If $ \bigcup N_\alpha $ is stationary, then $ \{ \min(N_\alpha) \}$ is stationary

It's the last set-theoretic question for tonight, I promise. I'm trying to figure out why the following holds true: Suppose we have a regular, uncountable cardinal $ \kappa $ and a disjoint family ...
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2answers
24 views

Unboundedness of the set of subgroups of a cardinal

I'm trying to figure out why the following is true: Let $ \kappa $ be an uncountable, regular cardinal. Suppose we turn it into a group (i.e. there are operations $ (\cdot, ^{-1}, e) $ with which $ ...
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1answer
43 views

Checking whether some sets are clubs in $ \aleph_2 $ and $ \aleph_1 $

I'm getting acquainted with the stationary sets and clubs. I don't yet quite get everything well, so I'd appreciate some help with this question: which of the following sets are clubs, contain a ...
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1answer
20 views

Prove that a specific subset $A$ of a nontrivial vector space $V$ over an infinite field $\mathbb{F}$ is infinite

Let $V$ be a nontrivial vector space over an infinite field $\mathbb{F}$. Suppose $V = \bigcup\limits_{i=1}^{n} S_i$, where $S_i$ is a proper subspace of $V$. We assume that $S_1$ is not included in ...
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Comparing infinite cardinals [closed]

I have a question concerning infinite cardinals which I found on an old exam paper: Let $c=2^{\aleph_0}$, $x=2^c, y=2^{2^c}, z=2^{2^{2^c}}$. Put $x^{y^z}, x^{z^y}, y^{z^x}$ in ascending order. I'm ...
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0answers
29 views

How does the cardinality of the set of all functions from $A$ to itself relate to that of $A$?

If $A$ is a set with cardinality $c$, what can we say about the cardinality of the set of all functions from $A$ to itself?
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2answers
20 views

Is this proof about equicardinality correct and/or rigorous? Can it be helped?

Here's the proof than a Cartesian product of two countable sets is countable(the proof is used, for example, in C.Pugh's "Real Mathematical Analysis" with one exception: they prove equicardinality of ...
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1answer
31 views

Connected Linear Graph not Path-Connected

Given a set of vertices $\{x_\alpha\}$ whose cardinality exceeds $\aleph_1$, (assume the axiom of choice) connect each vertex with its successor by an edge, forming a linear graph. Choose two vertices ...
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2answers
28 views

Sum of cardinals of all intersections: elegant alternative proofs?

I once read the following problem: compute $$\sum_{A,B\in\mathcal{P}(\Omega)}\operatorname{card}(A\cap B)$$ where $\Omega$ is a set of cardinal $n>0$ and $\mathcal{P}(\Omega)$ the set of the sets ...
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3answers
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What is an example of two sets which cannot be compared?

In set theory, if we do not assume the Axiom of Choice, we cannot prove the Trichotomy Law between cardinals. That is, we cannot prove that for any two sets, there exists an injection from one to the ...
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3answers
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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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Prove that the set of all infinite subsets of $\mathbb{N}$ is uncountable.

For this problem, my proof was: If we want to express out the set of all the finite subsets, $F$. $F = \{\{n_{1}\},\{n_{1},n_{2}\},\{n_{1},n_{2},n_{3}\},\cdots\}$ with $n_{1} \in \mathbb{N}$, $n_{2} ...
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30 views

Can the unit interval map bijectively to a region?

The Hilbert Curve shows that there exists a surjection from the unit interval to the unit square. I was wondering, does there exist a bijection from the unit interval to the unit square?
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1answer
50 views

Cardinality of Power set of naturals equal to $\Bbb{N}^\Bbb{N}$

The question: Decide with proof which has greater Cardinality $\Bbb{N}^\Bbb{N}$ or $2^\Bbb{N}$. My intuition: They will be the same. By Cantors argument and the continuum hypothesis, both will have ...
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1answer
65 views

Bijection from $\mathbb {Z}^3$ to $\mathbb {Z}$

I am not a mathematician. Let $\mathbb {Z}$ be a positive integer set. I need to know whether there exist a bijection from $\mathbb {Z}^3$ to $\mathbb {Z}$, what might be a possible mapping? I know ...
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1answer
28 views

Why is this set club?

I am currently reading a proof on properties of stationary sets and one step of the proof does not make a whole lot of sense to me. The proof asserts that If $\kappa$ is a regular cardinal and ...
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1answer
28 views

Maximal Sets and Bijections

I'm struggling with this question (The function $f(x) = x^2 -3$): Let $A = \{x \in R : x \geq 0\}$. Determine a maximum set $B$ such that $f : A \rightarrow B$ is a bijection. Let $g : B \rightarrow ...
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1answer
77 views

Can the cardinality of a strictly ordered set exceed the cardinality of the natural numbers?

I'm putting some thought into the CH at the moment and a proof of the answer to this question would be most helpful if anybody would be so kind as to help me out: Can the cardinality of a strictly ...
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72 views

Why can't you count up to aleph null?

Recently I learned about the infinite cardinal $\aleph_0$, and stumbled upon a seeming contradiction. Here are my assumptions based on what I learned: $\aleph_0$ is the cardinality of the natural ...
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7answers
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Why do we classify infinities in so many symbols and ideas?

I recently watched a video about different infinities. That there is $\aleph_0$, then $\omega, \omega+1, \ldots 2\omega, \ldots, \omega^2, \ldots, \omega^\omega, \varepsilon_0, \aleph_1, \omega_1, ...
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1answer
89 views

Ultraproduct with no long descending sequence

I have a countably infinite well-ordered structure $M$ (over a countable language if it helps), and an uncountable regular cardinal $κ$, and I wanted to construct an elementarily equivalent structure ...
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1answer
52 views

If $p_n$ is the $n^{th}$ prime, is it ever appropriate to speak of $p_{\aleph_0}$?

If $p_n$ is the $n^{th}$ prime, is it ever appropriate to speak of $p_{\aleph_0}$? I'm no math student. Your pardon if this is just some clearly obvious and easy answer, I'm just not seeing it. ...
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1answer
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Well-ordering of sets of cardinal numbers

Proposition For every cardinal number $m$ there is a definite next larger cardinal number. This proposition is proved on page 136 of "Proofs from the Book" using the fact that any set of ordinal ...
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“Proof” that $\text{cof}(\omega_\lambda)<\omega_\lambda$ if $\lambda$ is a nonzero limit ordinal

The following lemma is from this introduction to cardinals. Lemma 2.7. Let $\omega_\alpha$ be a limit cardinal. Then $\alpha$ is a limit ordinal and $\text{cof}(\omega_\alpha)=\text{cof}(\alpha)$. ...
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1answer
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For every cardinal $\kappa$, $\kappa^+$ is regular

Again I'm struggling with a proof from this introduction to cardinals. Lemma 2.6. For every cardinal $\kappa$, $\kappa^+$ is regular. Proof. If not, then there would be a cofinal map ...
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1answer
36 views

Using the axiom of choice to choose bijections

I couldn't think of a better question title. I am trying to understand the proof of theorem 1.8 in this introduction to cardinals. Theorem 1.8. Let $\kappa\in CARD$. Let ...
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2answers
25 views

Validity of certain arguments about the countability of infinite sets

I am trying to get an understanding, in layman's terms / on an intuitive level, why some arguments about the countability of infinite sets are valid, and some arguments which seem almost identical on ...
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1answer
183 views

Showing a cardinal is regular

I've been thinking about this question, but to no avail and I've got to ask. How to show that for $\kappa\geq\aleph_0,$ $\mu=\min\{\lambda: \kappa^{\lambda} > \kappa\}$ is regular? If I wanted a ...
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2answers
34 views

Another characterization of the cofinality?

Is it true that $cf(\kappa)=\min \{\lambda:\ \kappa^{\lambda}>\kappa\}$? $cf(\kappa)$ is certainly $\geq$ than that minimum since $\kappa^{cf(\kappa)}>\kappa$, but I don't know how to tackle ...
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2answers
84 views

What are some applications of large cardinals?

Most mathematicians don't often encounter cardinalities larger than that of the continuum, it seems? What are some results outside of pure set theory/logic that rely on the properties of larger ...
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Equivalence Relations and Cardinality

I'm looking at the question below from a past paper: What is an equivalence relation? Say that two sets $X$ and $Y$ are related via the relation $\rho$ if $X$ and $Y$ have the same cardinality. Prove ...
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1answer
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Without the Axiom of Choice, $\aleph_0<2^{\aleph_0}$ implies $\aleph_1\le 2^{\aleph_0}$?

Question: In ZF (so AC does not necessarily hold) does the following claim hold? $\aleph_0<2^{\aleph_0}$ implies $\aleph_1\le 2^{\aleph_0}$ This question arose to me when reading the top ...
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1answer
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Comparability of a set and a subset of power set.

It's well known that for any set $A$, $A < P(A)$. But now, I have some question that, WITHOUT AC, can we guarantee that $A \leq X$ or $X \leq A$ whenever $X \subseteq P(A)$? Thank you.
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1answer
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Equinumerousity of operations on cardinal numbers

I want to prove for all Cardinal numbers $a$, $b$, $c$ that: $(a \cdot b)^c =_c a^c \cdot b^c$ $a^{(b+c)} =_c a^b \cdot a^c$ $(a^b)^c =_c a^{b \cdot c}$ I know that for 1. it's enough to show ...
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3answers
55 views

Question about $\aleph$-fixed point

I am working through a proof on cardinals I found and can't reason some of the steps. The proposition is that there is an $\aleph$-fixed point, i.e. there is an ordinal $\alpha$ (which is ...
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1answer
97 views

Regarding the axiom $2^\kappa = 2^{\kappa^+}$ for regular cardinals $\kappa$, and its relationship to a couple of other axioms.

(Take ZFC as background.) The following two statements both follow from GCH: ICF. Injective continuum function. The continuum function (i.e. $\kappa \mapsto 2^\kappa)$ is injective. ...
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What is $n^{\aleph_0},n\in\mathbb N$

Can I say that $$n^{\aleph_0}=2^{\aleph_0\log_2n}=2^{\aleph_0}=\aleph_1$$
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1answer
65 views

Does $\operatorname{card}(X) < \operatorname{card}(Y)$ imply $\operatorname{card}(X^2) < \operatorname{card}(Y^2)$ without choice?

I looked to see if this question was already posted, but did not find anything. Please let me know if this is a duplicate. Assume $X, Y$ are infinite sets such that there is an injection $X \to Y$ ...
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1answer
37 views

Number of subsets with the same cardinality

Suppose we have a set $S$ with cardinal number $n$, such that $n+n = n$. Consider the set, T, of all subsets with cardinality $n$. How can I show that the cardinality of $T$ is $2^n$? (Without the ...