3
votes
1answer
83 views

$\aleph_\omega$ and large ordinals?

Assuming the GCH, each time you take the powerset of an aleph number, the subscript increases by one. So there is $\aleph_0, \aleph_1, \aleph_2\ldots \aleph_\omega\ldots \aleph_{\epsilon_0}\ldots$ and ...
0
votes
1answer
31 views

How to show $A_1 \approx A_2 \iff \mathrm{card}(A_1)=\mathrm{card}(A_2)$

For any set $A_1$ and $A_2$, let us define the relation $\approx$ if there exists a bijection between $A_1$ and $A_2$. Then I want to show that $A_1 \approx A_2 \iff ...
5
votes
0answers
57 views

Raising a partial function to the power of an ordinal

Consider a set $X$, and let $f : X \rightarrow X$ denote a partial function. Then for natural $n$, we can define $f^n$ as iterated composition, e.g. $f^2 = f \circ f$. Now suppose that $X$ is also ...
0
votes
1answer
40 views

Are these ordinals cardinals?

Consider $\omega_2 \times \omega$ and $\omega \times \omega_2$ with ordinal arithmetic. Then $| \omega_2 \times \omega | =\omega_2$ and $| \omega \times \omega_2 | =\omega_2$ Does this imply ...
15
votes
5answers
1k views

Why study cardinals, ordinals and the like?

Why is the study of infinite cardinals, ordinals and the like so prevalent in set theory and logic? What's so interesting about infinite cardinals beyond $\aleph _0 $ and $\mathfrak{c} $? It seems ...
1
vote
2answers
48 views

Is a limit ordinal necessarily a cardinal?

Maybe this is a trivial question. I see that every infinite cardinal is necessarily a limit ordinal, but is the converse true ?
1
vote
1answer
43 views

Is the cardinality of an ordinal, $\alpha$ less than $\alpha$?

Let $k^+=\{\alpha : \alpha \hspace{2mm} \text{is an ordinal,and} \hspace{2mm} |\alpha|<\kappa\}$ where $k^+$ is an infinite cardinal. I am trying to understand the proof the the following ...
0
votes
1answer
23 views

An alternative succinct proof needed for trivial cardinality fact

Let $|X|$ denote the cardinality of a set, i.e. the least ordinal $\alpha$ such that there is a bijection between X and $\alpha$. For any sets $X$ and $Y$ we write $X\preccurlyeq Y$ if the exists an ...
1
vote
1answer
43 views

Cardinality of Orderings of $\mathbb{R}$

For a finite set $S$ there are $\vert S\vert!$ orderings of its elements. What is the cardinality of all orderings of $\mathbb{N}$? What would $$\vert \mathbb{N}\vert!$$ mean? Is it ...
1
vote
1answer
95 views

Ordinality of a Set

What is the difference between Ordinal number and cardinal number of a set?....I have a confusion in understanding the difference between the two.Can anyone help me to understand these two things? ...
1
vote
1answer
53 views

Prove: $2^{\aleph_0}\not=\aleph_{\epsilon_0}$

Prove: $2^{\aleph_0}\not=\aleph_{\epsilon_0}$ I would like a hint for this problem
2
votes
1answer
36 views

Set of Cardinals

Let $A$ be a set of cardinals. Prove that there is a cardinal that that is greater than every cardinal in $A$. Assume that there isn't such a cardinal. Then for any cardinal $x$ there is $y\in A$ such ...
0
votes
2answers
94 views

Some questions about elementary set theory (cardinal and ordinal numbers)

I have three questions about elementary set teory and i don't figure out how to solve them: 1)Let $X$ a subset of the cardinal number $2^{\aleph_0}$ (seen as an initial ordinal). Is true or false ...
4
votes
1answer
113 views

Bijection between countable ordinals and reals

The set of all countable ordinals is $\omega_1$, which has a cardinality of $\aleph_1$. When accepting the continuum hypothesis, $2^{\aleph_0} = \aleph_1$, so a bijection between countable ordinals ...
2
votes
1answer
109 views

Godel's pairing function and proving c = c*c for aleph cardinals

I have a few questions about Godel's pairing function and proving that c = c * c for aleph cardinals. Mostly, though, I'm concerned that most of the proofs I've seen are erroneous, and this concerns ...
1
vote
3answers
91 views

Why $\omega+1$ and $\omega^2$ are not cardinal numbers?

I see the following definition of cardinal number in notes: An ordinal $\alpha$ is a cardinal number if $|\beta|<|\alpha|$ for all $\beta\in\alpha$. Why $\omega+1$ and $\omega^2$ are not cardinal ...
1
vote
2answers
78 views

For all infinite cardinals $\kappa, \ (\kappa \times \kappa, <_{cw}) \cong (\kappa, \in).$

I don't understand the proof to the a/m claim. How we know that $\eta < \kappa$ and $(\alpha,\beta)<_{cw} (0, \eta) $ and hence $h: \mu \to \eta \times \eta$ is injective? Appreciate if anyone ...
2
votes
3answers
46 views

about definition of a cardinal

Definition: the cardinality of a set $A, |A|$ is the least ordinal s.t. $A \sim \alpha$ Definition: We define a cardinal to be an ordinal $\alpha$ s.t. $\alpha = |\alpha|.$ i.e, an ordinal s.th. ...
3
votes
2answers
152 views

Two exercises on set theory about Cantor's equation and the von Neumann hierarchy

Good evening to all. I have two exercises I tried to resolve without a rigorous success: Is it true or false that if $\kappa$ is a non-numerable cardinal number then $\omega^\kappa = \kappa$, where ...
4
votes
1answer
293 views

Cardinal Arithmetic versus Ordinal Arithmetic

I am reading Philosophy, not Set Theory, so please excuse the naivety of my question. My question concerns the wildly different character of ordinal arithmetic versus cardinal arithmetic. The ...
14
votes
2answers
219 views

The preorder of countable order types

Consider the set $\mathcal{O}$ of order types corresponding to all posets of cardinality at most $\aleph_0$. The set $\mathcal{O}$ is a preorder under embeddability of its elements (note that some ...
6
votes
0answers
111 views

AC iff $P(\delta)$ can be well-ordered for all $\delta\in{\bf On}$ [duplicate]

I'm trying to do the following exercise: Exercise. Assume ZF. Then AC is equivalent to the statement that $P(\delta)$ can be well-ordered for all $\delta\in{\bf On}$. I'm struggling with the ...
2
votes
2answers
125 views

I read that ordinal numbers relate to length, while cardinal numbers relate to size. How do 'length' and 'size' differ?

I read that ordinal numbers relate to length, while cardinal numbers relate to size. How do 'length' and 'size' differ? Note : I am an absolute novice, and I'm having a little trouble visualizing ...
0
votes
1answer
116 views

The limit elements of well-ordered proper classes

For any well-ordered class $W$ with no maximum element, we can define a successor function $f_W : W \rightarrow W$ by asserting that $f_W(w) = \mathrm{min}\{x > w \mid x \in W\}.$ This allows us to ...
-3
votes
1answer
77 views

Statements regarding ordinal numbers [duplicate]

Let $m$ and $n$ be infinite ordinal numbers Which of the following is true a) $m<n \Rightarrow |m|^{|m|}<n^{|n|}$ b) $m+n$= Max{$m,n$} c) $m=n \Rightarrow |m|=|n|$ d)$|m|=|n| \Rightarrow ...
2
votes
1answer
82 views

Are the closed ordinals (apart from $0$ and $1$) precisely the regular cardinals?

Given a partially ordered set $P$, a collection of partially ordered sets, call it $\mathcal{Q}$, and an arbitrary function $f : P \rightarrow \mathcal{Q},$ we can form a new partially ordered set ...
1
vote
1answer
117 views

Cardinal numbers

Suppose $m, n$ are infinite ordinal numbers. $$a) m=n → |m|=|n|$$ $$b)|m|=|n| →m=n$$ $$c)m<n→ |m|<|n|$$ $$d)|\max{(m,n)}|< |m|+|n|$$ $$e)|m|<|n| →|m|^{|n|}<|n|^m$$ Which of the above ...
0
votes
2answers
171 views

Closed, unbounded subset of a cardinal.

I missed two lectures in my set theory course, and now I don't understand the homework problems. One is this: let $\kappa$ be a regular uncountable cardinal. Show that the following sets are closed ...
2
votes
1answer
169 views

Can we embed the ordinal and cardinal number systems into larger, more convenient systems of arithmetic?

We can embed $\mathbb{N}$ in a larger number system, such as $\mathbb{Z}$, $\mathbb{Q}$ or $\mathbb{R}$, for convenience. Now since $\mathbb{N}$ is extended by $\mathrm{Ord}$ and $\mathrm{Card}$, the ...
6
votes
2answers
194 views

Confusion about cofinality

I'm confused about the notion of the cofinality of a cardinal. Since I think the source of the confusion is the Von Neumann cardinal assignment, my first question is: Question 0. Is there an article ...
4
votes
1answer
63 views

If $cf(\kappa)=\lambda$, then is every sequence of length $\lambda$ cofinal in $\kappa$?

Take $\omega_1$ for instance. Let's say I have a sequence of (distinct) ordinals of length $\omega_1$. Will this sequence be cofinal in $\omega_1$?
1
vote
2answers
84 views

Cardinal numbers are identified with the set of ordinals preceding them

Here is a description of cardinal number. Cardinal numbers are identified with the set of ordinals preceding them. Is this OK?
0
votes
0answers
54 views

Can we conclude that $|\prod_{\xi \in Ord, \xi=1}^{\xi<\alpha}\xi|=2^{|\alpha|}$ for all infinite ordinal $\alpha$ in $\mathsf{ZFC}$?

According to this question, $\prod_{\kappa \in Crd, \kappa=1}^{\kappa<\aleph_\alpha}\kappa$, the cardinal product of all positive cardinals smaller than a given infinite cardinal $\aleph_\alpha$ ...
1
vote
2answers
183 views

Is the class of all ordinals independent of set theory?

I am little confused with this. In any model of set theory (or is only in ZFC?), we start with an initial set, and then by transfinite recursion we build the universe of sets, which is a proper class. ...
1
vote
1answer
128 views

It is possible to generalize the “real” line to be able to embed $\omega_1$ or any uncountable ordinal into a finite segment of it?

This question is motivated from a previous question, but is in itself independent of it. So, I understand that it is not possible to embed $\omega_1$ or any uncountable ordinal into the real line, ...
3
votes
1answer
75 views

How to show this space $X$ is countably compact, first countable?

Consider the subspace $X$ of $(2^\omega)^+$, i.e., the smallest cardinal greater then $2^\omega$, equipped with the ordered topology consisting of all ordinals of countable cofinality. How to ...
7
votes
1answer
115 views

Why should the union of such a family of sets have cardinality $\aleph_1?$

Let $\mathcal A$ be a family of infinite countable sets which is linearly ordered by inclusion and such that $\bigcup\mathcal A$ is uncountable. I have to prove that $|\bigcup\mathcal A|=\aleph_1.$ I ...
4
votes
3answers
244 views

The real line has cardinality at most $\aleph_2$, but transfinite ordinal space has arbitrarily high cardinality: what is wrong?

In the context of supertasks, people and mathematicians are comfortable with the idea of transfinite ordinal time, that is, that time can be divided into an arbitrarily high number of steps. In most ...
5
votes
2answers
408 views

Why are infinite cardinals limit ordinals?

My book states this as obvious, but it isn't so trivial to me. thanks
2
votes
1answer
89 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 ...
4
votes
2answers
93 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
76 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
126 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 ...
3
votes
1answer
73 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
4answers
320 views

What's the definition of $\omega$?

This is a follow up on a comment to one of my previous questions. What's the definition of $\omega$? Are the following equivalent definition of $\omega$: $\omega$ is the initial ordinal of ...
2
votes
2answers
129 views

Possible inaccuracy in Wikipedia article about initial ordinals

I quote from the Wikipedia article: "So (assuming the axiom of choice) we identify $\omega_\alpha$ with $\aleph_\alpha$, except that the notation $\aleph_\alpha$ is used for writing cardinals, and ...
3
votes
3answers
310 views

Which set is unwell-orderable?

In textbook it says that every well-orderable set is equipotent to an initial ordinal number. However, of course unwell-orderable cannot equipotent to any ordinal number, but is there any set is ...
5
votes
3answers
212 views

Question on a proof from Jech's Set Theory of: If $X$ is a set of cardinals, then $\sup X$ is a cardinal.

The proof: Let $\alpha= \sup X$. If $f$ is a bijective mapping of $\alpha$ onto some $\beta < \alpha$, let $\kappa \in X$ be such that $\beta < \kappa \le \alpha$. Then $|\kappa|=|\{f(\xi): \xi ...
1
vote
3answers
274 views

How to prove that the set of all countable ordinals, $\omega_1$, is uncountable / has the cardinality $\aleph_1$?

The only reasoning I've seen given for this is that it's uncountable because it can't include itself an element. I'm a little unconvinced and was looking for a more proper formal proof demonstrating ...
0
votes
1answer
175 views

If $\kappa$ is a cardinal, $\aleph$ is any $\aleph$-number, and if $\kappa\leq\aleph$ then $\kappa$ can be well ordered as well.

I'm having trouble understanding the statement: If $\kappa$ is a cardinal, $\aleph$ is any $\aleph$-number, and if $\kappa\leq\aleph$ then $\kappa$ can be well ordered as well. I understand the ...