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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$|X|=|X\cup\{a\}|?$

Let $X$ be an infinite set and $a\notin X$. Prove $$|X|=|X\cup\{a\}|$$ This is so intuitively obvious but upon inspection it appears quite non-obvious. How might one prove this? Do I need the axiom ...
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1answer
13 views

Cardinal Inequality without using Choice

Let $k$ be an infinite cardinal (i.e. not in bijection with any natural number), show that $\mathbb{N} \leq 2^{2^k}$ This is very easy with choice, without it I don't even know where to start.
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2answers
112 views

Neighborhood chain in $\mathbb{R}^\mathbb{R}$

Edited after SamM's comment: Consider the topological space $\mathbb{R}^\mathbb{R}$, with the usual topology. Pick a point $x \in \mathbb{R}^\mathbb{R}$ and a neighborhood $V = V_0$ of $x$. I wish to ...
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2answers
78 views

Can the extended real number $+\infty$ be compared to transfinite numbers such as $\aleph_0$?

If not, why not? If so, is ∞ greater than or less than $\aleph_0$? Edit: the discussion in comments (including comments on a deleted answer) have made me think that the best way to put the issue is ...
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2answers
74 views

Does $1-\frac{1}{2^\omega}$ equal 1? What about $1-\frac{1}{2^{\aleph_0}}$?

(Correct if I'm wrong on any of this.) Recently, I've been learning about transfinite ordinals and cardinals. For some cases, I understand the difference between ordinals and cardinals, for instance ...
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1answer
34 views

Why $|A \cup B \cup C| = |A| + |B| + |C|$ in this case? Linear algebra, the dimension of subspaces and their sum

I have a trouble understanding a certain proof from Roman's textbook "Advanced Linear Algebra" that For subspaces $S$ and $T$ of a vector space $V$ we have $\dim{S} + \dim{T} = \dim(S+T) + \...
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3answers
193 views

Fixed points and cardinal exponentiation

Let the function $F: On \rightarrow On$ be defined by the following recursion: $F(0) = \aleph_0$ $F(\alpha+1) = 2^{F(\alpha)}$ (cardinal exponentiation) $F(\lambda) = \sup\{F(\alpha):\alpha \in \...
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0answers
79 views

Prove that Real Numbers and the interval (-∞,0) have the same cardinality. [closed]

1.Prove that Real Numbers and the interval (-∞,0) have the same cardinality. 2.Prove that Real Numbers and the interval (-∞,0] have the same cardinality. How could I solve this two by using ...
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1answer
32 views

Cardinality of the set of $2\times 2$ real matrices with determinant $1$ [closed]

Please, I need the cardinality of the set of $2\times 2$ real matrices with determinant $1$. I have no idea where to start.
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0answers
54 views

How to solve probability when sample space is infinite?

I came up with a random problem yesterday: Suppose that in a random trial, each point $(x,y)$ where $x,y \in \mathbb{R}$ and $0 \leq x,y \leq 1$ is assigned a value of $0$ with 50% chance and a ...
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1answer
20 views

Cardinality of set of functions with coefficients from a set with cardinality omega

Let $A_1$, and $A_2$ be subsets of $\mathbb{R}$ of cardinality $\aleph_0$, $\aleph_1$ respectively. Let $P_1$ be the set of polynomials of the form $a_n(x^n) + a_{n−1}(x^{n−1}) +··· a_1x + a_0$ where $...
4
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1answer
62 views

Is $2^{2^{\aleph_0}}$ a higher cardinality than $2^{\aleph_0}$?

As far as I understand, $2^{\aleph_0}$ is the cardinality of the real numbers (and whether this equals $\aleph_1$ is the continuum hypothesis). But would $2^{2^{\aleph_0}}$ be of a higher cardinality ...
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0answers
29 views

The usual bijection between $[0,1]$ and $[0,1]\times[0,1]$

While trying to explain to someone else how you can have a bijection between $[0,1]$ and $[0,1]\times[0,1]$, I found an issue in the usual bijection that we use. The usual bijection that I'm talking ...
2
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1answer
29 views

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 \lambda\...
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1answer
76 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
23 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
79 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 ...
3
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2answers
43 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
21 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 $\kappa_j\...
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3answers
65 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
17 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
38 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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0answers
58 views

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? (or also: for profinite groups?) I suspect that the theory of ...
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1answer
39 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
26 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 $ \...
2
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1answer
50 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 club,...
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1answer
22 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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1answer
43 views

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
22 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
1k views

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
2k views

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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2answers
49 views

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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2answers
33 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
62 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 ...
3
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1answer
70 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
30 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
29 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
88 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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2answers
87 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
3k views

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
92 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 ...
3
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1answer
53 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
83 views

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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2answers
38 views

“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
46 views

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 $f:\...
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1answer
37 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 $X=\bigcup_{\alpha<\...