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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Help understanding cardinal multiplication and infinite Cartesian products

The cardinal product of two sets is defined to be the cardinality of the Cartesian product. The Cartesian product is: $$\prod_{\alpha \lt\beta}\kappa_{\alpha}=\{f\mid f\colon\beta\rightarrow ...
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A generalization of “any countable limit ordinal is the union of a sequence of increasing ordinal”

Using the fact that every countable ordinal is isomorphic to a closed subset of $\mathbb Q$, I find out that any countable limit ordinal is the union of a sequence of increasing ordinal. Now I'm ...
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176 views

Is $2^{\aleph_0} = \aleph_1$?

I was reading a thread on Examples of Common False Beliefs in Mathematics on MathOverflow, in which a user wrote: $$2^{\aleph_0} = \aleph_1$$ This is a pet peeve of mine, I'm always surprised ...
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1answer
29 views

uncountable repetitions

I have a question (or two) about recursive naming conventions. Consider the following recursive naming sequence: Base step: Let S be any nonempty set. Let x be any arbitrary element of S. Let S* be ...
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1answer
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What is the limit of the cardinality of a set of bins in finite range, as bin width approaches zero?

Let's say that we divide the region $(0,1)$ into $N$ bins of width $1/N$. Of course, it makes sense to take the limit $1/N \rightarrow 0$ in this configuration, because that's simply how we define an ...
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1answer
24 views

Is $\operatorname{card}(I)=\operatorname{card}(D)$

When I was answering number of integrable functions is greater than number of differentiable functions I got to wonder if the inequality was strict. So with $\mathcal I$ being the set of integrable ...
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Cantor's Diagonal: Why not a 1-2 Correspondence between the Naturals and Reals?

Hopefully I'm following Cantor's Diagonal Argument with a minimum of distortion and omission: We start from an enumeration T of all infinite binary sequences. We then construct a list S of elements ...
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2answers
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Cardinality of $\{ (x, y) \in \mathbb{R}^2 \mid \left| x \right| + \left| y \right| = 1 \}$ and $\{ (x, y) \in \mathbb{R}^2 \mid x^2 + y^2 = 1 \}$

Do $\{ (x, y) \in \mathbb{R}^2 \mid \left| x \right| + \left| y \right| = 1 \}$ and $\{ (x, y) \in \mathbb{R}^2 \mid x^2 + y^2 = 1 \}$ have the same cardinality? One can draw a square in the two ...
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1answer
95 views

What's the largest number

Originally this question started as 'what is the largest number' using $\aleph_0$ as a start, and continuing using concepts such as ${\aleph_0}^{\aleph_0}$, and Knuth's Tower notation $\uparrow$, so ...
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2answers
50 views

$\forall \alpha \exists \beta: \beta > \alpha$ where $\alpha$ and $\beta$ cardinals

I have to prove ZF $\vdash$ $\forall \alpha \exists \beta:\beta > \alpha$, where $\alpha, \beta-$ cardinal numbers. I can prove it only in ZFC. Let's fix some cardinal number $\alpha$. By ...
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1answer
22 views

$\sigma$-algebra with cardinality $\aleph_0$ [duplicate]

Can a $\sigma$-algebra in a set $X$ have cardinality $\aleph_0$, the cardinality of the naturals? I do not have a clue on how to start with this? Can someone please give me a hint?
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1answer
60 views

Why does this author define cardinality indirectly?

I'm studying Enderton's Elements of Set Theory and in the page 129 he defines what it means two sets being equinumerous: After that in the page 136 he defines cardinality: Why doesn't he define ...
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1answer
68 views

Is the cofinality function monotonic?

Is the cofinality function $\operatorname{cf}$ monotonic? I.e., if $\lambda \le \kappa$ for cardinals $\lambda$ and $\kappa$, does it then follow that $\operatorname{cf}(\lambda) \le ...
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0answers
30 views

Mathematics with and without continuum hypothesis

This is a follow-up to a recent question. Are there "interesting" differences between CH-mathematics and (non-CH)-mathematics? Has anybody developed mathematics with c = $\aleph_2$? $\aleph_3$? ... ...
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2answers
50 views

Sets of “Isolated” Cardinals

Let $C\neq\emptyset$ be a set of infinite cardinals with the property that NO member of $C$ occurs as the supremum of strictly smaller members of $C$. So the cardinals in $C$ are sort of "isolated". ...
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The cardinality of dense subsets of infinite-dimensional Hilbert spaces

If $H$ is an infinite-dimentional Hilbert space, then does $\dim H$ coincide with the smallest cardinal of a dense subset of $H$?
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3answers
279 views

A question about cardinal number.

Let $X$ be a infinite set and $n$ be a positive integer. We denote the cardinal number of $X$ by $|X|$ and denote the family of all subsets of X which contains n elements by $\mathfrak{F}$. Then ...
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1answer
20 views

Is ordering of (possibly infinite) sets by cardinality a total ordering?

Given sets $A$ and $B$. Can you show that either there exists an injective map of $A$ into $B$ (that is, a map such that each element of $A$ maps to an element of $B$ and no two elements of $A$ map to ...
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3answers
71 views

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
43 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
80 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
43 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
45 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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2answers
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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
31 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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3answers
70 views

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
26 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
66 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
82 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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Cardinality of symmetric density functions relative to the cardinality of all density functions

Is there anyone who has some idea bout the following question? $X=$(total number of all pairs of probability density functions $(f_0,f_1)$ on the real numbers) and let $Y=$(total number of all ...
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3answers
195 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
40 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
23 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
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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
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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
35 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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2answers
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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
49 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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22 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
24 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
47 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
31 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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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
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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 $\alpha > |A|$ , ...
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1answer
60 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
63 views

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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63 views

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
57 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
39 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 ...