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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Decidability of the cardinality of a set given that the Continuum Hypothesis is independent from ZFC

I'm a Total Amateur (TM), please forgive me if this question makes no sense. The Continuum Hypothesis states that there are no sets with cardinality strictly between that of the integers and the ...
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1answer
46 views

Cardinality of sets of reals without choice

Assuming just ZF (no axiom of choice): Does $\aleph_n\leq|\mathbb{R}|$ for all $n<\omega$ imply $\aleph_\omega\leq|\mathbb{R}|$? (with $\kappa\leq|\mathbb{R}|$ meaning that there is a set of reals ...
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3answers
91 views

Teaching cardinality

I would like to give a class of 60 minutes to my undergraduate students about cardinality. I would like to begin with the definition of cardinality and end with one or two good application of this ...
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0answers
55 views

How to calculate the dimension of an infinite direct product of copies of a field?

Let $F$ be a field and $I$ an arbitrary infinite index set. I'd like to know how to calculate the dimension of $\prod_{i\in I}F$. By the way, I know $\dim(\prod_{i\in I}F)\geqslant ...
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1answer
47 views

Can you say that almost all $\mathcal{C}^\infty$ functions are not polynomials?

This question asked whether there are functions other than the trigonometric ones whose Maclaurin series contains infinitely many terms, i.e. that never become zero under repeated differentiation. ...
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How can I prove that the cadinality of a set minus a finite number of elements of it is still the same as the original set?

A is a finite subset of S, which is an infinite set. How can I prove that $|S| = |S \setminus A|$? I just finished proving that $|T \cup S|$ where $T$ is infinite and $S$ is countable is $|T|$. They ...
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1answer
38 views

How could I prove that the cardinality of the union of two sets is equal to R? $|T U S| = |T| = |\mathbb{R}|$

I have to prove that $|T \cup S|$ where $T$ is infinite and $S$ is countable, equal to $|T|$, and this is also $|\mathbb{R}|$. How can I approach this? $|T \cup S| = |T| = |\mathbb{R}|$ I tried to ...
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What is the cardinality of the set of all functions from $\mathbb{Z} \to \mathbb{Z}$?

How can I approach this? I have to find the cardinality of the set of the functions from $\mathbb{Z} \to \mathbb{Z}$ and I have no idea on how to solve it. Can someone hint me here? The approach ...
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0answers
28 views

What mean $L(\mathbb{R})$ and $L(\mathbb{R})^*$?

I found them relating a cardinality question here. Does it have anything to do with regularity/computability?
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1answer
82 views

What axioms are needed in proofs of the independence of the continuum hypothesis?

My understanding is that the proofs that CH and not-CH are consistent with ZFC are both about ZFC and in ZFC. Is it possible to do these proofs about ZFC but in a weaker axiomatic system? (It is also ...
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1answer
99 views

Measure of an elementary set in terms of cardinality

In Terry Tao's textbook on measure theory and integration, he notes that, given an elementary set $A$, the length of $A$, denoted $|A|$, may be written discretely as $$|A| = \lim_{n \to ...
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21 views

What is the Cardinality of all symmetric density function pairs on reals?

$X=$(total number of all pairs of probability density functions $(f_0,f_1)$ on the real numbers) and let $Y=$(total number of all symmetric probability density functions $(f_0,f_1)$ on the real ...
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3answers
284 views

Bijection between open and closed interval [duplicate]

I am not sure how to approach the following problem: Show the open interval $(a,b)$ is bijective with the closed interval $[c,d]$. I was thinking of using $a+u$ where $u$ is a really small number ...
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1answer
46 views

cardinality of polynomial

What is the cardinality of the following sets? (Choose from finite, countably infinite, or uncountably infinite.) The set of polynomials of the form $ax+b$ with $a \in\Bbb N$ and $b \in\{0,1\}$ ...
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1answer
24 views

Finite Sets Proof on Domains [duplicate]

Just wanted some help with this little proof.: Let X and Y be Finite Sets. Prove that |X^Y| = |X|^|Y|
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1answer
44 views

Proof of the definition of cardinal exponentiation [duplicate]

I really cannot seem to get my head around the definition of cardinal exponentiation with regards to finite sets: $|X|^{|Y|}=|X^Y|$ How would one even begin to prove this? Isn't $X^Y$ the set of all ...
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1answer
82 views

Help with intuition on Cardinal Arithmetic Problems

It happens a lot to me that when I find an intuitive model (picture) of a mathematical entity, the proofs left as exercises in books are very easy to solve. For example when dealing with filters and ...
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1answer
38 views

Do homeomorphic metric spaces have equal minimal cardinality of dense subsets? [closed]

Let $X,Y$ be two homeomorphic topological spaces and let $d(X)$ denote the minimal cardinality of a subset $A \subseteq X$ such that $\bar A=X$, i.e., $A$ is dense in $X$. Then is it true that ...
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Let $A, |A|=a$ be a set where $a$ is infinite. How many equivalence relations are there over $A$?

Let us denote the set of equivalence relations $B$. So, the first direction is to say that the number of equivalent relations won't exceed the number of relations, that is $|P(A\times A)|=2^a$. Now, ...
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1answer
57 views

On the cardinality of $\mathbb R \times …\aleph_1 {times}$ and $\mathbb R \times …2^{\aleph_0} \space {times}$

I think I can prove that closure of every countable set in any metric space has cardinality at most $\mathcal c=2^{\aleph _0}$ . So if a metric space is separable i.e. has a countable dense subset $A$ ...
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37 views

Finite sum over uncountable set

Consider the sum $S=\sum_{x\in I}P(x)$, where $P(x)$ are positive real numbers. When the index set $I$ is finite, $S$ is of course finite. When $I$ is countably infinite, it is also possible that $S$ ...
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1answer
27 views

For each ordinal $\alpha$, $\alpha\le \aleph_{\alpha}$

This property is mentioned in http://en.wikipedia.org/wiki/Aleph_number I cannot find a contradiction assuming otherwise. Maybe this is proved by transfinite induction? $\aleph_\alpha$ is defined ...
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1answer
59 views

Replacing an ordinal with its cardinality in a partition relation

In The Higher Infinite, Kanamori claims that if $\alpha$ is a cardinal, and $\beta \to (\alpha)^\gamma_\delta$ for some $\beta$, then the least such $\beta$ is a cardinal. I can't seem to think of a ...
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1answer
38 views

Countable union of sets of cardinality $c$ has cardinality $c$

The book Theory of Functions of a Real Variable by I. P. Natanson, proves that a denumerable or finite union of pairwise disjoint sets of cardinality $c$ has cardinality $c$. The proofs given in the ...
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1answer
42 views

Question related to ordinal number without using Axiom of Choice.

Can we proof this result without using Axiom of Choice :- $$A\cap \alpha=\emptyset \,\,\,\, \mbox{and}\, \, \, A\times \alpha \sim A\cup \alpha$$ then there is an $A^{'} \subset A$ such that $\alpha ...
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Clarification on the proof of Theorem 8. 11 (Hungerford)

If $\alpha$ and $\beta$ are cardinal numbers such that $0\neq \beta \leq \alpha$ and $\alpha$ is infinite, then $\alpha\beta=\alpha.$ Sketch: Let $A$ be an infinite set with $|A|=\alpha$ and let ...
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1answer
17 views

Let $A$ be any non-empty set and $\alpha$ be an infinite cardinal . When can we say $|A^\alpha|=\alpha$ ?

Let $A$ be any non-empty set and $\alpha$ be an infinite cardinal . When can we say that the cardinality of $A^\alpha$ ($A \times A\times ...$ $\alpha$ times ) is $\alpha$ ? When $\alpha > |A|$ , ...
4
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1answer
63 views

Does $\lambda^2 \leq \kappa^2 \Rightarrow \lambda \leq \kappa$ imply the axiom of choice?

I'm aware that the statement "for all cardinals $\kappa$, $\kappa^2 = \kappa$" is equivalent to the axiom of choice (I believe this was proved by Tarski). More generally, does anyone know if the ...
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3answers
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Let $V,W$ be two countably infinite dimensional vector space over the same field , then are $V,W$ isomorphic as vector spaces?

Let $V,W$ be two countably infinite dimensional vector space over the same field , then are $V,W$ isomorphic as vector spaces ? And please give example of two non-isomorphic uncountable dimensional ...
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1answer
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Enderton's Elements of Set Theory Scott's Trick Exercise (page 207 problem 31)

31. Define kard $A$ to be the collection of all sets $B$ such that (i) $A$ is equinumerous to $B$, and (ii) nothing of rank less than rank $B$ is equinumerous to $B$. (a) Show that kard $A$ is ...
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2answers
75 views

Halmos' Naive Set Theory Cardinal Arithmetic Exercise

On page 95 of Halmos' Naive Set Theory, we get the exercise If $\{a_i\}$ and $\{b_i\}$ are families of cardinal numbers such that $a_i< b_i$, then $$\sum_i a_i<\prod_ib_i$$ I know that we ...
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1answer
40 views

$\operatorname{Card}(X) \leq\operatorname{Card}(Y)$ iff $\aleph (X) \leq \aleph(Y)$

For any two sets $X$ and $Y$, we write $\operatorname{Card}(X)\leq\operatorname{Card}(Y)$ if an injection $X \rightarrow Y$ exist. I have tried Suppose $\aleph (X) \leq \aleph(Y)$, where $\aleph ...
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3answers
53 views

Set cardinality, function onto, open unit square maps into real number set

I have a question in my homework that I have trouble solving it. I'm not sure if I understand the question actually. I'll attach the question below and hope someone could give me any hints. Consider ...
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40 views

A question regarding Silver's Theorem

I am reading Jech's book Introduction to Set Theory. As an introduction to Silver's theorem, it is stated that: If $\aleph_\alpha$ is a strong limit cardinal and if $k < \aleph_\alpha$ and ...
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1answer
43 views

How to prove $cf(\aleph_{\omega_1})=\omega_1$?

I am reading Jech's & Hrbacek's book "Introduction to set theory" and this is question 2.2 from chaper 9: Prove $cf(\aleph_{\omega_1})=\omega_1$ Isn't it clear from the fact that ...
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1answer
91 views

How do I show the existence of a weakly inaccessible cardinal is not provable in ZFC?

So I know that if we assume the existence of inaccessible cardinals than we can show ZFC is consistent. How do I show that the existence of such cardinals is not provable in ZFC?
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determine the cardinality of $\{C \subseteq \mathbb N \space|\space \mathbb N - C \text{ is finite}\}$ [duplicate]

what is the cardinality of this set : $\{C \subseteq \mathbb N \mid \mathbb N - C \text{ is finite }\}$ So it must mean that $C$ is infinite, but even though its infinite we know how ...
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1answer
84 views

Why we use $\mathrm{Fn}(\kappa\times\lambda,2,\lambda)$ to force $2^\lambda\ge\kappa$ instead of $\mathrm{Fn}(\kappa\times \lambda,2)$

I am reading Kunen's Set Theory and I learn that, $\operatorname{Fn}(\kappa\times\omega,2)$ forces $2^{\aleph_0}\ge|\kappa|$, where $$\operatorname{Fn}(I,J,\lambda)=\{p:p\text{ ...
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1answer
136 views

ZFC + not-CH, can a set with cardinality between $\aleph_0$ and $2^{\aleph_0}$ be defined?

Continuum hypothesis states, there is no set with cardinality between the integers and the reals. There is a milestone result, that CH is independent from ZFC. That means, both of ZFC + CH, and ZFC + ...
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1answer
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How to determine cardinality of an infinite set using Aleph numbers?

So I was reading a little bit about cardinal infinities, and I thought it was pretty interesting. However I wanted to know a little bit more about how to use them. For example, how would I determine ...
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1answer
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Powers of $\mathfrak{c}^+$

Denote by $\mathfrak{c}^+$ the cardinal successor of continuum. Can we prove in $\mathsf{ZFC}$ that $(\mathfrak{c}^+)^{\aleph_0} = \mathfrak{c}^+$? I guess not. Of course this question is ...
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1answer
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If $f\colon\kappa\rightarrow\kappa$, then the set of all $\alpha < \kappa$ such that $f(\xi) < \alpha$ for all $\xi < \alpha$ is closed and unbounded.

If $f\colon\kappa\rightarrow \kappa$, then the set of all $\alpha < \kappa$ such that $f(\xi) < \alpha$ for all $\xi < \alpha$ is closed and unbounded. This is from Jech's book (page 103) so ...
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Ordinals that are not cardinals [duplicate]

I am reading Jech's set theory and he defines a cardinal number as an ordinal $\alpha$ (a cardinal) if $|\alpha| \neq |\beta|$ for all $\beta < \alpha$, and he says that all infinite cardinals are ...
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1answer
25 views

bijective, one-to-one, and number of elements

How does one reconcile the following (seemingly) contradiction in using "number of elements" argument? In the "range" [0,1] in R there are more points than in N, to be shown as "take the inverse of ...
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2answers
38 views

Cardinality of the set of all complex sequences converging to zero.

I was asked to show that the set of all complex sequences converging to zero has the same cardinality as the set [0,1]. This is the only hole in a proof that I am working on. I need to show there ...
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1answer
100 views

A club-guessing exercise

I came across this club-guessing exercise on Cardinal Arithmetic by Abraham and Magidor in the Handbook of Set Theory. Let $\kappa, \lambda$ be regular cardinals $\kappa^{++}<\lambda$ and let ...
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1answer
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Question on Komjáth's “three clouds may cover the plane”

I am reading a wonderful paper by Komjáth, "Three clouds may cover the plane," and am having difficulty proving that certain sets are countable. Assume CH (the continuum hypothesis) holds. Let ...
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59 views

Vector spaces over Z/2Z,countably many predicates added,invariant

Suppose we have a class of vector spaces over $Z/2Z$ with predicates $P_n,n\in N$ added for independent subspaces of co-dimension 2. (BTW, independent in which sense,$P_n$ is independent with ...
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1answer
44 views

Does any set admit a total order? [duplicate]

Is it true that any set $P$ can be endowed with a total order $"\leq" \subseteq P\times P$?
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39 views

Prove there is either a chain or an antichain of infinite cardinal.

Let $K$ be an set of infinite cardinal, $X$. Let $(K,\le)$ be a partially ordered set. Prove there is either a chain $C$ such that $|C|=X$ or there is an antichain $A$ such that $|A|=X$. I guess I ...