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.

learn more… | top users | synonyms (1)

0
votes
0answers
11 views

Supremum of a regular cardinal.

Let $k$ be a regular uncountable cardinal; that is, $k$ cannot be written as the sum of less than $k$ cardinals each with less than $k$ elements. Is it necessarly true that $\sup k = k$? (that is, ...
0
votes
0answers
17 views

A question on product space [duplicate]

If $|X|=\mathfrak c$, then what is the cardinality of the product space $X^{\omega}$? Thanks very much.
2
votes
3answers
63 views

What properties does $A\to B$ satisfy under 1-1 correspondence?

A 1-1 correspondence between two sets $A$ and $B$ is a function $f\colon A \to B$ satisfying what properties? I do know that we say that two sets $A$ and $B$ are equivalent, and we write $A \sim B$ ...
1
vote
3answers
24 views

Limit Ordinals as Infinite Ordinals and other questions

I am studying set theory and I am confused in the following: Are limit ordinals the same as infinite ordinals? I would say yes since the least non-zero limit ordinal is $\omega$. Infinite limit ...
0
votes
3answers
82 views

Proving $a + a = a$ if a statement is true

That's the question : Let $a$ be a cardinality such that this following statment is true : For every $A, C$, if $ A \subseteq C$, $|A| = a$ and $|C| > a$, then $|C \setminus A| > |A|$. ...
0
votes
2answers
22 views

What is the cardinality of all binary sequnces (infinte and finite) that the sequnce 01 does not apper in them

As the title suggests, the question is : What is the cardinality of all binary sequnces (infinte and finite) that the sequnce 01 does not apper in them ? I'll tell you where im stuck, let's say f is ...
0
votes
1answer
21 views

Cardinality and Set Thoery

(|x| = cardinal # of x for clarification) let A,B be two finite sets, show that $|A \cup B| = |A| + |B| -|A\cap B|$ Proof: let $x\in A\cup B$ $x \in (A -A\cap B) + (B- A\cap B) + (A \cap B)$ let ...
0
votes
0answers
26 views

Show that same cardinal number defines and equivalence relation

Same cardinal number must satisfy all three properties : symmetry, reflexivity, transitivity Symmetry Suppose bijection $f:A→A$, Then by definition, |A| = |A| Reflexivity Suppose bijections ...
1
vote
2answers
21 views

Relation between successor cardinals and power sets

What are the known relation between successor cardinals $\kappa^+$ and power sets $2^\kappa$ (when GCH is not assumed)? For example, is it true that $\kappa^+ \le 2^\kappa \le \kappa^{++}$? In ...
1
vote
1answer
19 views

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 ...
3
votes
1answer
40 views

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 ...
11
votes
4answers
197 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 ...
0
votes
1answer
35 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 ...
-2
votes
1answer
22 views

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 ...
3
votes
1answer
33 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 ...
-2
votes
1answer
69 views

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 ...
1
vote
2answers
31 views

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 ...
0
votes
1answer
97 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 ...
2
votes
2answers
52 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 ...
0
votes
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?
1
vote
1answer
61 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 ...
0
votes
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 ...
1
vote
0answers
32 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 = $\aleph_2$? $\aleph_3$? ... ...
1
vote
2answers
52 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". ...
-2
votes
0answers
31 views

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 $H$?
2
votes
3answers
280 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 ...
2
votes
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 ...
1
vote
3answers
72 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 ...
4
votes
1answer
44 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 ...
2
votes
3answers
82 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 ...
1
vote
0answers
44 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 ...
1
vote
1answer
46 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. ...
4
votes
2answers
44 views

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 ...
0
votes
1answer
32 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 ...
1
vote
3answers
71 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 ...
2
votes
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?
2
votes
1answer
75 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 ...
3
votes
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 ...
0
votes
0answers
19 views

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 ...
1
vote
3answers
203 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 ...
-1
votes
1answer
41 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 $b \in\{0,1\}$ ...
0
votes
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|
2
votes
1answer
39 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 ...
3
votes
1answer
76 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 ...
0
votes
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 ...
1
vote
2answers
39 views

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, ...
1
vote
1answer
50 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 $A$ ...
0
votes
0answers
23 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 $S$ ...
1
vote
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 ...
3
votes
1answer
50 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 ...