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.

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18 views

What is the cardinality of this 7-tuple-coordinate set?

Due to 150 letters rule, therefore I have to write the question here. What is the cardinality of $\{ ( a_1 , a_2 , ... , a_7 ) | \{ a_1 , a_2 , ... , a_7 \} = \{ 1, 2, 3, 4, 5, 6, 7 \} \land \sum_{...
7
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1answer
107 views

Does a free algebra over a nontrivial monad have a well-defined dimension?

Let $(T,\mu,\eta)$ be a nontrivial monad on $\mathbf{Set}$. By nontrivial here I mean there is $X$ with $|T(X)|>1$. Suppose that $TX \cong TY$ as $T$-algebras (both with the usual free $T$-algebra ...
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2answers
60 views

Guide to cardinals and ordinals

I'm looking for a short textbook concerning cardinal and ordinal numbers (their definition and arithmetic). It may be rather formal or complicated, but shall contain the necessary methods for a ...
3
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1answer
28 views

Long sequences in singular cardinals can't be cofinal

My attempts to understand a solution to the Kunen exercise (two well orderings on a cardinal $\kappa$ necessarily agree on a set of size $\kappa$) have foundered on the following claim. If someone ...
2
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4answers
99 views

Prove: there exists 3 sets: $A, B, C \subseteq \mathbb{N}$ such that: $A\cap B\cap C =\emptyset$ and $|A|=|B|=|C|=\aleph_0$?

Prove: there exists 3 sets: $A, B, C \subseteq \mathbb{N}$ such that: $A\cap B= B \cap C = A \cap C = A\cap B\cap C = \emptyset$ and $|A|=|B|=|C|=\aleph_0$? also, the sets must exists: $$|\mathbb{N} ...
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0answers
46 views

An infinite cardinal number into a power of a finite cardinal number

In "Naive Set Theory" Halmos gives an exercise: Prove that if $a, b$ are cardinal numbers, $a$ is infinite, $b$ is finite, then $a^b = a$ I struggle to use the known properties of cardinal numbers ...
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1answer
104 views

Different sets of glyphs for aleph-numbers

So I know the list of lists of aleph-numbers that goes: $\aleph_0, \aleph_1, \aleph_2, \aleph_3 ...$ $\aleph_{\aleph_0}, \aleph_{\aleph_1}, \aleph_{\aleph_2}, \aleph_{\aleph_3} ...$ $\aleph_{\...
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0answers
31 views

Two well-orderings of an infinite cardinal agree on a large set (Part II)

This is a follow-up to my earlier question on the Kunen exercise, which asks for a proof that any two well-orderings on an infinite cardinal $\kappa$ agree on a set of size $\kappa$. I'm perfectly ...
3
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0answers
74 views

Consistency of $\mathfrak{b}<\mathfrak{s}$

I'm reading a paper written by Vera Fischer and Juris Steprans related with cardinal invariants of the continuum where they obtain, using finite support iteration of c.c.c partial orders, a model ...
0
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2answers
39 views

$\mid{B}\mid=\aleph_0 , A\cap{B}=\emptyset$. Prove $\mid{A} \cup B \mid\,=\,\mid{A} \mid$ [duplicate]

Let $A$ be an infinite set , and a set $B$. $\mid{B}\mid=\aleph_0 , A\cap{B}=\emptyset$. Prove $\mid{A} \cup B \mid\,=\,\mid{A} \mid$ . What I thought to do is to take a set $C\subseteq A\, ,\, \mid{...
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4answers
118 views

how to prove that: $|X| = 2^{\aleph_{0}}$, given $X = \{ A\in P(\mathbb{N}) \vert \ |A^{c}| = \aleph_{0} \}$

Let $X = \{ A\in P(\mathbb{N}) | \ |A^{c}| = \aleph_{0} \}$ Prove\ Disprove that: $|X| = \aleph_{0}$ My attempt: I'd like to disprove that: $|X|=\aleph_{0}$, By proving that $|X|=2^{\aleph_{0}}$. ...
3
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1answer
103 views

Visualisation of ordinal vs cardinal arithmetic

I came around a nice article talking about large countable ordinals with examples such as: Let's have a book with $\omega$ pages (each 1/2 the thickness of previous so we can fit them in a book of ...
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1answer
34 views

Set of disjoint circles of radius $1$ on the plane countable?

Give is a set of disjoint circles of radius $1$ on the plane. Is this set countable ? I don't think so. In my opinion, the set with only one circle with radius 1in the plane is also uncountable, isn't ...
9
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1answer
97 views

Are cardinals with the same “continuum function” equal?

Let $\kappa, \lambda$ be two infinite cardinals such that for all infinite $\mu, \mu^\kappa = \mu^\lambda$. Is it the case that $\kappa =\lambda$ ? First of all, clearly if the generalized continuum ...
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0answers
29 views

An elementary question about cardinal arithmetic without AC [duplicate]

Suppose $|X|=2^{2^{\aleph_0}}$, $A \subset X$ and $|A|=2^{\aleph_0}$. Is it possible to prove that $|X \setminus A|=2^{2^{\aleph_0}}$ without using the axiom of choice and its consequences?
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0answers
59 views

unknown existence of bijection

Here on the page 4 in the proof of fact 2.6(a) , I do not follow why there is a bijection $$g:\lambda\to\cal P_{\aleph_0}(p)$$ so that we can fix it. We only require $$|A|\leq\lambda,$$ so $\cal P_{\...
2
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1answer
61 views

A generalization of the concept of finiteness

For concreteness' sake, let the underlying set theory be ZFC. A set $x$ will be called connex (I've adapted the terminology from here) iff for every $y, z \in x$ either $y \subseteq z$ or $z \...
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1answer
43 views

filter notation

I've come across this notation $$\cal D^+,$$ where $\cal D$ is a filter over some cardinal $\lambda.$ If I remember correctly they said that this is the set of all positive sets, but I cannot ...
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1answer
57 views

Prove $\mid\mathbb{N}\mid \leq \, \mid \mathbb{N}^{\mathbb{N}} \mid$

I need to prove that $$\mid\mathbb{N}\mid \leq \, \mid \mathbb{N}^{\mathbb{N}} \mid$$ I assume that I need to show that there is function $ f: \mathbb{N} \rightarrow \mathbb{N}^{\mathbb{N}}$ which is ...
1
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1answer
25 views

ideal $\check I(\lambda)$: the definition is missing

Here on the page 15 in the definition 1.14 is $\check I[\lambda]$ defined but $\check I(\lambda)$ ideal not. Does someone know what is this round brackets notation? EDIT BTW, on the page 9 there in ...
1
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1answer
47 views

Prove that set is stationary.

In Thomas Jech's 'Set Theory' there is the following statement: For a regular uncountable cardinal κ and a regular λ < κ the set $\{\alpha < \kappa: cf(\alpha) = \lambda\}$ is stationary. But ...
0
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1answer
38 views

True cofinality tcf

Please, here on the page 6, in the definition 0.11 I wonder what happens with the definition of "tcf" when $I$ is this poset: o o \ / \ / o This $I$ ...
0
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1answer
56 views

Ordinal and cardinal arithmetic

For ordinal exponentiations can we write ? $$ω<ω^ω<ω^{ω^ω}<\cdots$$ I know, if $\omega$ is the first infinite cardinal,we have power set, which is For example, $\aleph_0<{\aleph_0}^{\...
8
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1answer
281 views

Is cardinality continuous?

Let the underlying set theory be ZFC. Let $x_1 \subseteq x_2 \subseteq \dots$ and $y_1 \subseteq y_2 \subseteq \dots$ be ascending sequences of sets such that, for every $n \in \{1,2,\dots\}$, $|x_n| =...
0
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2answers
52 views

Given a set $X$ and a cardinal $\kappa$ such that $\kappa<|X|$, can I always find a subset of $X$ which has cardinality $\kappa$?

Given a set $X$ and a cardinal $\kappa$ such that $\kappa<|X|$, can I always find a subset of $X$ which has cardinality $\kappa$? If that's the case, then there must be a subset of the reals which ...
2
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2answers
78 views

Do I understand the continuum hypothesis correctly?

I read this concept new. The purpose of this question is: I want to check if I understand this concept correctly or not. What I understand: We can not create such a set, consisting of ...
0
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2answers
33 views

It is true that the criterio of comparation of sets is equivalent to Axiom choice? [duplicate]

We say that two sets $A$ and $B$ are comparables if and only if $|A|\leq |B|$ or $|B|\leq |A|$. I want prove that this criterio is equivalent to axiom of choice. If I use Zorn lemma it is simple ...
0
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1answer
78 views

Proof checking about cardinality of all Real numbers

Cardinality of real numbers in the interval $[0,1]$ equals to $2^{\aleph_0}$. Now, I want to show that cardinality of all real numbers is equal to cardinality of real numbers in the interval $[0,1]$. ...
0
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1answer
23 views

Under which conditions does the non-computable infinite sequences have a computable subsequences?

Under which conditions does the non-computable infinite sequences have a computable subsequences? And if we subtract non-computable infinite sequences which has a computable subsequence, from non-...
1
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2answers
55 views

On Ways to Show that the Cardinalities of $(0,1)$ and $\mathbb{R}$ are the Same?

"Everyone knows" that to show that $\mathbb{R}$ is uncountably infinite, it suffices to show that the real numbers in the interval $(0,1)$ cannot be listed, which can be accomplished by Cantor's ...
0
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1answer
60 views

A question about the cardinality of $\theta$-closed hull of a set

I have been reading a proof for the following proposition Proposition: Let $X$ be a Urysohn space. If $A$ is a subset of $X$, then $|[A]_\theta|\leq |A|^{\chi(X)}$ Here, $[A]_\theta$ denotes the $\...
0
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1answer
48 views

A question about $2^{\aleph_0}$

Is the cardinality of the infinite sequences consisting of $\left\{0,1\right\}$ or $\left\{0,1,2\right\}$ or $\left\{0,1,2,3,\cdots,9 \right\}$ elements equal to the cardinality of the infinite ...
1
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1answer
48 views

Is there a cardinal function such that the intersection of a family of open sets with the same cardinality has an open intersection? [duplicate]

I am working in some cardinal functions in topological spaces, and I need to either define a new cardinal function or use an existing one that behaves in the following way: Lets say $\varphi$ is my ...
0
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1answer
25 views

Cardinality of the continuum and binary expansion of numbers

The accepted answer to this question regarding the continuum hypothesis contains the sentence: "I am assuming you know that |ℝ|=2^(ℵ0), which can be proven by looking at binary expansions of numbers ...
1
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1answer
44 views

What is the cardinality of $L^{\infty}(\mathbb{R})$?

I know of this answer for $L^p(\mathbb{R})$ where $p < \infty$: What is the cardinality of $L^p(\mathbb R)$, $1 \le p < \infty$? But, what about $L^{\infty}(\mathbb{R})$, can we prove any ...
0
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1answer
34 views

Show that the cardinality of ($X$ ∪ {$x$}) is equal to the cardianlity of ($X$)+$1$

Let $X$ be a finite set, and let $x$ be an object which is not an element of $X$. Then $X$ ∪ {$x$} is finite and #($X$ ∪ {$x$}) = #($X$)+$1$. Note that #-here means cardinality. Suppose the cardinality ...
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1answer
74 views

Example of non well-orderable set in ZF. [duplicate]

Definition: For a set $x$ define its cardinality by $|x|=\min\{\alpha\in On\mid\alpha \approx x\}$. where $On$ is the calss of all ordinals, $\alpha\approx x$ means there is a bijection $f:\alpha\...
1
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1answer
79 views

Let $S$ be the set of all possible functions mapping $\{\sqrt 2, \sqrt 3, \sqrt 5, \sqrt 7 \}$ to $\Bbb Q$, find the cardinality of $S$.

Let $S$ be the set of all possible functions mapping $\{\sqrt 2, \sqrt 3, \sqrt 5, \sqrt 7 \}$ to $\Bbb Q$, find the cardinality of $S$. At first I wanted to use the theorem that for any sets $A$ and ...
2
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2answers
75 views

Prove that for any set A, if $\vert \Bbb N \times \Bbb R \vert \geqslant \vert A \vert$, then $\vert \Bbb R \vert \geqslant \vert A \vert$.

Prove that for any set A, if $\vert \Bbb N \times \Bbb R \vert \geqslant \vert A \vert$, then $\vert \Bbb R \vert \geqslant \vert A \vert$. I am considering proving $\vert \Bbb N \times \Bbb R \vert =...
1
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0answers
74 views

Prove that for any set A and B, the cardinality of the set of all functions mapping A to B is $\vert B \vert ^ {\vert A \vert}$

What I do is for finite sets A, B, let $A={a_1, a_2, ...a_n}$ and $B={b_1, b_2, ...b_m}$ A function f assigns each element $a_i$ of $A$ to an element $b_j = f (a_i)$ of $B$; there are $m$ ...
0
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0answers
52 views

Infinite Time Turing Machine and Hypertask

Good Day, I would like to ask this question. Infinite Time Turing Machine ordinals are countable ordinals. Also I have read that ITTM either halts or repeats itselve after countably many steps. Does ...
6
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1answer
73 views

On the distributive number $\mathfrak h$

The distributive number $\mathfrak h$ is defined as the least cardinal $\kappa$ such that there exists a family of $\kappa$ open dense subsets in the preordered set $([\omega]^\omega,\subset^*)$ whose ...
1
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1answer
40 views

Does $|A|^{|A|}=2^{|A|}$ hold for any infinite set $A$? [duplicate]

Because I know the cardinality of all the functions$f:[0,1]\to \mathbb{R}$ is $2^{c}$(c is the cardinality of continuum).I wonder whether this holds generally.
4
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1answer
111 views

If $X$ is an infinite set, then is $\sum_{i=1}^\infty |X|=|X|$?

I used the above to solve a problem, but I am not sure if this is true (and if it is true does it require a proof or is it obvious). I think it is true because $|X| + |X| = |X|$, so it should follow ...
0
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4answers
91 views

A Set is Infinite if, and only if, it is in One-to-one Correspondence with a Proper Subset of Itself

Can someone explain what that means? How can there exist an injective function from an infinite set to a proper subset of itself. A function from a set A to a set B where B has fewer elements than A ...
2
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2answers
46 views

Proof Verification that Every Finite Set has a Unique Cardinality

Is my proof correct for proving that every finite set has a unique cardinality? My part of the proof is as follows: "Let $A = \{a_1, a_2, ..., a_n\}$ be an arbitrary set with $n$ elements, and let $f:...
0
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2answers
91 views

How can I describe the set of homomorphisms and what is its cardinality?

There are actually two questions here. Neither of these is a homework question. Let $\text{hom}(\mathfrak{A},\mathfrak{B})$ (not to be confused with the Hom functor) be the class$^0$ of homomorphisms ...
3
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1answer
48 views

Proof that every infinite set $X$ has an uncountable number of topologies on it.

I am currently self-studying topology without tears. I am working on the following question: 1.2.7(iii): (open sets) If $X$ is an infinite set of cardinality $\aleph$, prove that there are at ...
5
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0answers
97 views

Set of Ordinal Functions with singular cofinality

Is there a set $A$ of regular cardinals such that the partial order $(\prod A, <)$ has singular cofinality? Here $\prod A$ is the set of all ordinal functions on $A$ with $f(\kappa)<\kappa$ for ...
4
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2answers
61 views

Regarding the Post: “Does $k+\aleph_0=\mathfrak{c}$ imply $k=\mathfrak{c}$ without the Axiom of Choice?”

I have been searching around for a proof (that I can understand) that the cardinality of the irrationals equals $\mathfrak{c},$ the cardinality of the reals. (This, of course, is different than ...