Cardinality is a notion of size for sets, usually denoted by $|A|$ as the "cardinality of $A$". With finite sets the cardinality is simply the number of elements which are members of a set. Dealing with infinite sets we can measure them in different ways. Cardinal numbers are very natural in the ...
1
vote
2answers
69 views
Computing $\kappa^{<\lambda}$, for cardinals $\kappa$ and $\lambda$
I'm trying to show that, for $\lambda$ an infinite cardinal and $\kappa$ any cardinal, that $$\kappa^{<\lambda} = \sup\{\kappa^\theta:\theta<\lambda\land\theta\text{ a cardinal}\},$$ where ...
4
votes
3answers
289 views
cardinality of set of all real continuous functions
Could somebody explain to me how to prove that the cardinality of all real continuous functions is $c$ ?
The first problem is that I don't know how to show that each real continuous function $f: X ...
0
votes
3answers
176 views
Proof of equal cardinality $|\Bbb N \times\Bbb N \times\Bbb N| = |\Bbb N|$
How do I prove that the following sets have equal cardinality?
$|\Bbb N \times\Bbb N \times\Bbb N| = |\Bbb N|$ ($|\Bbb N \times\Bbb N| = |\Bbb N|$ also for that matter)
$|\Bbb Z \times\Bbb Z| = ...
3
votes
3answers
228 views
Easiest way to prove that $2^{\aleph_0} = c$
$\aleph_0$ is the cardinality of the set of natural numbers, $\aleph_0 = |N|$. $c$ is the cardinality of the continuum, i.e. the set of real numbers $c = |R|$.
I know that $|P(A)| = 2^{|A|}$. This ...
4
votes
2answers
79 views
Weak cardinal powers and singular cardinals
Suppose $\kappa > \operatorname{cf}(\kappa)$. Show that:
i) if $\kappa$ strong limit then $\kappa^{<\kappa} = \kappa^{\operatorname{cf}(\kappa)}$
ii) if $\kappa$ not strong limit then ...
15
votes
3answers
431 views
For any two sets $A,B$ , $|A|\leq|B|$ or $|B|\leq|A|$
Let $A,B$ be any two sets. I really think that the statement $|A|\leq|B|$ or $|B|\leq|A|$ is true. Formally:
$$\forall A\forall B[\,|A|\leq|B| \lor\ |B|\leq|A|\,]$$
If this statement is true, ...
2
votes
1answer
51 views
Possible typo in Just/Weese's set theory
In Just Weese on page 197 there are the following corollaries:
Regarding Corollary 24: Is this a typo and should say "$CON(ZF) \not\rightarrow CON(ZF + \exists \text{ "a strongly ...
1
vote
2answers
61 views
Number of Vertices of Graphs
So, I was looking at some graph theoretical stuff, more specifically Topological Graph Theory, and I had a question about the definition of graphs: is there usually a condition in the definition ...
1
vote
1answer
77 views
Assuming $(GCH)$, strongly inaccessible and weakly inaccessible coincide
My book says
"... If $GCH$ holds, then the notions of strongly inaccessible and weakly inaccessible cardinals coincide, ..."
In $ZFC$ I can prove this. But the paragraph from which I have excerpted ...
1
vote
2answers
64 views
Defining strong limit cardinals in $ZF$
I do not understand the following passage/footnote in the book I am currently reading:
An initial ordinal $\lambda$ is called a strong limit cardinal if
$2^\kappa < \lambda$ for every $\kappa ...
0
votes
1answer
72 views
Proving Axiom Schema of Replacement holds in $H_\lambda$
I think I proved the following, can you tell me if my proof is correct?
Exercise 23: Show that if $\lambda$ is a regular infinite cardinal then $\langle H_\lambda , \overline{\in} \rangle$ satisfies ...
2
votes
2answers
79 views
Possible typo in proof of Bukovský-Hechler
In the proof of the following theorem:
Theorem 29 (Bukovský-Hechler): Let $\kappa, \lambda$ be infinite cardinals such that $\mathrm{cf}(\kappa) \le \lambda$ and
$\mathrm{cf}(\kappa) < ...
0
votes
0answers
58 views
A quick question about proof of Bukovský-Hechler
The following is an exercise in Just/Weese (page 179),
Question 1: can you tell me if I got it right? Thank you!
Question 2: Shouldn't it be equality rather than less equals in $\mu = \sum_{\alpha ...
4
votes
2answers
165 views
Cardinal arithmetic gone wrong?
I am trying to calculate $\kappa^\lambda = \aleph_{\omega_1}^{\aleph_0}$.
I know that if $\kappa$ is a limit cardinal and $0 < \lambda < \mathrm{cf}(\kappa)$ then $\kappa^{\lambda} = ...
5
votes
2answers
168 views
Is it viable to ask in an infinite set about the Cardinality?
Can you ask given an infinite set about its cardinality? Does an infinite set have a cardinality? So, for example, what would be the cardinality of $+\infty$?
1
vote
2answers
73 views
Cardinal sums over successor ordinals are equal?
Can you tell me if the following claim and subsequent proof are correct? Thanks.
Claim: If $\alpha = \delta + 1$ is an infinite successor ordinal then $\sum_{\xi < \alpha } \kappa_\xi = \sum_{\xi ...
2
votes
1answer
72 views
Question about a proof about singular cardinals
The following is a lemma in Just/Weese on page 179:
Lemma 17: Let $\kappa$ be an infinite cardinal. Then $\kappa$ is singular iff there exist an $\alpha < \kappa$ and a set of cardinals ...
2
votes
3answers
102 views
Defining cardinals without choice
According to Wikipedia if we assume AC we define a cardinals number to be an ordinal that is not in bijection with any smaller ordinal.
Without AC, one takes the cardinality of a set $X$ to be the ...
0
votes
1answer
56 views
Deducing $|B^A|+|B^A|=|B^A|$ from $|A|+|A|=|A|$,
How attacking this question?
Show that if $A$ and $B$ are sets such that $A$ is infinite, $|A|+|A|=|A|$, and $|B|\geq 2$, then $|B^A|+|B^A|=|B^A|$
0
votes
2answers
74 views
Cardinality calculation
How to simplify the following:
$$2^{\aleph_0}(\aleph_0+\aleph_0)^{2^{\aleph_0}}$$
Thank you for every help.
4
votes
2answers
51 views
Question about proof of Hessenberg: $\kappa \cdot \lambda = \lambda$
The following is a theorem:
(Hessenberg) Let $1 \le \kappa \le \lambda$ where $\lambda$ is an infinite cardinal. Then $\kappa \cdot \lambda = \lambda$.
The proof in the book proceeds by transfinite ...
1
vote
1answer
66 views
Constructing a bijection between $\xi$ and $\xi + 1$
I did the following exercise, can you tell me if I have it right, thank you (Just/Weese p 176):
Show that $|\xi + 1|$ is either finite or equal to $|\xi|$. (here $\xi$ is an ordinal)
By ...
2
votes
1answer
94 views
Proving properties of enumeration of infinite cardinals
I am doing the following exercise from Just/Weese:
where $F$ is defined as follows:
(a) I'm not sure whether the following passes as "convince myself": Apparently $F$ is defined for all ...
4
votes
2answers
70 views
About a proof of “$\bigcup A$ is a limit cardinal”
Assume that if $A$ is a set of cardinals such that $A$ contains no largest element and assume that we have shown that $\bigcup A$ is a cardinal. Now we want to show that $\bigcup A$ is a limit ...
7
votes
1answer
119 views
Bijection between $\mathbb{Z}\times\mathbb{Z}\times\dots$ and $\mathbb{R}$
Is there a bijection between $\mathbb{Z}\times\mathbb{Z}\times\dots$ for countably infinitely many $\mathbb{Z}$'s and $\mathbb{R}$? That is, is $\mathbb{Z}\times\mathbb{Z}\times\dots$, repeated ...
3
votes
6answers
167 views
Proving the uncountability of $[a,b]$ and $(a,b)$
I am trying to prove that $[a,b]$ and $(a,b)$ are uncountable for $a,b\in \mathbb{R}$. I looked up Rudin and I am not too inclined to read the chapter on topology, for his proof involves perfect ...
11
votes
2answers
233 views
What's the difference between saying that there is no cardinal between $\aleph_0$ and $\aleph_1$ as opposed to saying that…
... $\aleph_1$ is the immediate successor of $\aleph_0$?
I was reading the wiki article on $\aleph_1$ where a distinction is made between the two. If there's isn't a cardinal between $\aleph_1$ and ...
2
votes
1answer
77 views
Cardinality of all possible partitions of a infinite set
I suppose this problem should be a commonplace, but I only find this one, in which notations are kind of idiosyncratic, along with a glaring defect.
My question is where I can find a more formal ...
3
votes
2answers
51 views
Why $\kappa^{<\kappa}=2^{<\kappa}$, if $\kappa$ is a regular and limit cardinal?
On Page 60, Set Theory Jech(2006)
(Show that)if $\kappa$ is regular and limit, then $\kappa^{<\kappa}=2^{<\kappa}$.
It's not difficult to show that $\kappa^{<\kappa}\geq2^{<\kappa}$. ...
2
votes
1answer
70 views
Is $\kappa^\lambda=2^\lambda$($2 \le \kappa<\lambda$,$\lambda$ infinite) valid in set models of ZF?
Let $2 \le \kappa<\lambda$(both cardinal numbers), in which $\lambda$ is infinite. Then these formula as follows hold where in ZFC:
$\lambda+\kappa=\lambda$
$\lambda\cdot\kappa=\lambda$
...
2
votes
2answers
44 views
Does $\kappa^\lambda=\kappa$ imply $\mu^\lambda=\mu$ for all $\mu>\kappa$, given $\kappa$, $\mu$, $\lambda$ are cardinals?
This problem is originated from the experience that I was trying to prove 5.19 and 5.20 on Page 60 of Set Theory, Jech(2006).
It seems to be right with AC, but I don't know how to prove it.
1
vote
1answer
43 views
$A+\alpha\sim A$ when $\omega\le\alpha<h(A)$
I have been considering about $A+\alpha\sim A$ when $\omega\le\alpha<h(A)$, in which $h(A)$ is the Hartogs number of $A$.
If there is $\alpha$ s.t. $\omega\le\alpha<h(A)$, we can get $\alpha ...
1
vote
2answers
78 views
Is there any Dedekind-infinite set can be split to two smaller Dedekind-infinite sets?
I have been considering about $A+\alpha\sim A$ when $\omega\le\alpha<h(A)$, in which $h(A)$ is the Hartogs number of $A$.
If there is $\alpha$ s.t. $\omega\le\alpha<h(A)$, we can get $\alpha ...
2
votes
1answer
97 views
question on formulations of Generalized Cardinal Hypothesis and Singular Cardinal Hypothesis
I hope this is not a silly question(well, not too silly, I hope). After all, a relevent question at a deeper level is already out there, even though it seems the solution is missing.
Why not ...
1
vote
1answer
162 views
Does $\mathbb R^2$ contain more numbers than $\mathbb R^1$? [duplicate]
Possible Duplicate:
bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$
Do the real numbers and the complex numbers have the same cardinality?
Does $\mathbb R^2$ contain more numbers ...
4
votes
1answer
79 views
Is $\aleph_0$ the minimum infinite cardinal number in $ZF$?
$\aleph_0$ is the least infinite cardinal number in ZFC. However, without AC, not every set is well-ordered.
So is it consistent that a set is infinite but not $\ge \aleph_0$? In other words, is it ...
0
votes
2answers
38 views
Can a set(in ZFC) be well-ordered with any order type equipotent to it?
Let $D$ be a set with cardinality $\aleph_\alpha$, and give an ordinal number $\beta$ between $\omega_\alpha$ and $\omega_{\alpha+1}$, can $D$ be well-ordered with order type $\beta$?
4
votes
1answer
99 views
Is this theory $\kappa$-categorical when $\kappa$ is infinite and not $\le \aleph_0$?
Let $\mathcal L$ be a language with only a binary predicate $E$, and let $T$ be a theory of structures in which $E$ is an equivalence relation which partitions the structure into two infinite ...
2
votes
1answer
56 views
Why $2^\kappa=\kappa^{\operatorname{cf}{\kappa}}$, if $\kappa$ is a strong limit cardinal?
On Page 58, Set Theory, Thomas Jech(2006) states the following fact without details.
Another fact worth mentioning is:
If $\kappa$ is a strong limit cardinal, ...
1
vote
1answer
57 views
Cardinal Exponentiation $\lim_{\alpha\to\kappa} \alpha^\lambda$
On Page 57 of Jech's Set Theory, Lemma 5.19
If $\kappa$ is a limit cardinal, and $\lambda \geq \operatorname{cf}{\kappa}$, then $\kappa^\lambda = (\lim_{\alpha\to\kappa} ...
8
votes
3answers
114 views
Cardinality of the Union is less than the cardinality of the Cartesian product
Suppose I have two collections of sets, $\{A_i\}_{i \in I}$ and $\{B_i\}_{i \in I}$. It is given in the problem that $$\mathrm{card}(A_i) < \mathrm{card}(B_i) \text{ for all }i \in I $$
I want to ...
4
votes
3answers
296 views
Do the real numbers and the complex numbers have the same cardinality?
So it's easy to show that the rationals and the integers have the same size, using everyone's favorite spiral-around-the-grid.
Can the approach be extended to say that the set of complex numbers has ...
4
votes
1answer
91 views
Why continuum function isn't strictly increasing?
Is there any example that for cardinal numbers $\kappa < \lambda$, we have $2^\kappa = 2^\lambda$?
My guess is that it only depends on whether GCH holds. Is it true?
3
votes
3answers
92 views
How to understand $\operatorname{cf}(2^{\aleph_0}) > \aleph_0$
As a corollary of König's theorem, we have $\operatorname{cf}(2^{\aleph_0}) > \aleph_0$
.
On the other hand, we have $\operatorname{cf}(\aleph_\omega) = \aleph_0$.
Why the logic in the latter ...
2
votes
2answers
161 views
The cofinality of $\aleph_{\omega\cdot9+3}$
I am studying for a test and I was able to find the cofinality 3 of the 4 ones given, but am having a lot of trouble with the 4th.
the 3 first ones are:
...
2
votes
1answer
59 views
The product of the finite non-zero ordinals, and the product of the finite non-zero cardinals.
I am trying to study for a test I have on Set Theory, and after being given the practice test, I am having some big problems with simple things. One question I am having a lot of problems with is:
...
2
votes
2answers
74 views
Prove that $S$ is countably infinite
Suppose we have the set $S\subset\mathbb{ N \times N}$ where $\mathbb N$ is the set of positive integers $\{1, 2,\ldots\}$ with the property: $S = \{\langle m, n \rangle\mid m \leq n\}$. Suppose ...
2
votes
3answers
129 views
Why the principle of counting does not match with our common sense
Principle of counting says that
"the number of odd integers, which is the same as the number of even integers, is also the same as the number of integers overall."
This does not match with my ...
2
votes
1answer
168 views
Counterexamples to the continuum hypothesis
Assume the continuum hypothesis is false, and add that as an axiom to ZF set theory. How many cardinalities are between the rationals and the reals in this case? Only one? Infinitely many? Countably ...
5
votes
2answers
145 views
A question about the cardinality of the set of all the bijections from $M$ to itself
$M$ is a set.We denote the set of all the bijections from $M$ to itself by $K$.Is the cardinality of $K$ larger than the cardinality of $M$?
