In the ZF set theory ordinals are transitive sets which are well-ordered by $\in$. They are canonical representatives for well-orderings under order-isomorphism. In addition to the intriguing ordinal arithmetics, ordinals give a sturdy backbone to models of ZF and operate as a direct extension of ...

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20
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3answers
705 views

Conflicting definitions of “continuity” of ordinal-valued functions on the ordinals

I've encountered the following definition in Kunen, Levy, and other places: A function $\mathbf{F}:\mathbf{ON}\to\mathbf{ON}$ is continuous iff for every limit ordinal $\lambda$, we have ...
21
votes
6answers
727 views

Embedding ordinals in $\mathbb{Q}$

All countable ordinals are embeddable in $\mathbb{Q}$. For "small" countable ordinals, it is simple to do this explicitly. $\omega$ is trivial, $\omega+1$ can be e.g. done as $\{\frac{n}{n+1}:n\in ...
2
votes
3answers
294 views

A sum of ordinals and a bijection

Let $a=a_{i\in n}$ is an ordinal-indexed family of ordinals. I call a structured sum of $a$ an order isomorphism from the lexicographically ordered set $\{ \left( i ; x \right) \; | \; i \in n, x\in ...
3
votes
1answer
156 views

countable product of $\omega_1 + 1$

How can I prove that the topological countable product of $\omega_1 + 1$ (with the order topology) is the union of $\omega_1$ closed nowhere dense sets?
-3
votes
1answer
134 views

Projections of ordinal numbers [closed]

Let $p$ is a product of two non-zero ordinal numbers. Knowing $p$ we can restore each of these two ordinals, right? How can we denote each of these two ordinals (supposing we know the value of $p$)? ...
1
vote
1answer
80 views

Product of ordinals notation

How to denote product of ordinals: $\cdot$ or $\times$? I'm not sure which of these two multiplication symbols to use.
-1
votes
2answers
157 views

Order of ordinals

Let $R_i$ is an ordinal for every $i\in n$ (where $n$ is also an ordinal) and $S_{i,j}$ is an ordinal for every $i\in n$ and $j\in m_i$ (where $m_i$ is also an ordinal for every $i\in n$). What is a ...
0
votes
1answer
97 views

An invertible function on ordinals

Let $R=R_{i\in n}$ is a family of ordinals where $n$ is also an ordinal. Let $q_i(x) = \sum_{j \in i} R_j + x$ for every $x\in R_i$. Is $q_i$ an invertible function?
3
votes
3answers
512 views

Definition of cofinality

Let $\alpha$ be a limit ordinal. We define $\operatorname{cf}\alpha$ to be the least limit ordinal $\beta$ such that there is an increasing $\beta$-sequence $\langle \alpha_{\xi} \mid \xi < ...
2
votes
1answer
108 views

Definition of a $\gamma$-filtered ordinal

This definition is part of Hovey's book on model categories: Let $\gamma$ be a cardinal. An ordinal $\alpha$ is $\gamma$-filtered if it is a limit ordinal and, if $A\subseteq \alpha$ and $|A|\leq ...
2
votes
1answer
173 views

Well ordering and maximal principle

I have to prove that given any poset $(P,\preceq)$ there exists a chain $S$ such that it is maximal (meaning that if $S\subseteq S'$ then $S=S'$). The book contains a proof using the axiom of ...
5
votes
2answers
159 views

The set of fixed points is closed (ordinals)

I am given that $F$ is a definite unary operation on the ordinals which is normal. That is, $$x<y\Rightarrow F(x)<F(y)$$ and furthermore $$F(y)=\sup\{F(x): x<y\}$$ if $y$ is a limit ordinal. ...
7
votes
1answer
208 views

If $\alpha\leq \beta,\gamma\leq \delta$ then $\alpha+\gamma\leq \beta+\delta$

The title is one of my homework assigments, (and the above greek letters are ordinal numbers) but this is the first one containing inequalities, and I dont really know how to do it. My guess is ...
6
votes
1answer
165 views

What are these operations on ordinals, analogous to multichoose, called?

Given well-ordered sets $\alpha$ and $\beta$, define $\left(\!\!{\alpha\choose \beta}\!\!\right)$ to be the set of weakly decreasing functions from $\beta$ to $\alpha$, ordered lexicographically; this ...
3
votes
2answers
371 views

Is the supremum of an ordinal the next ordinal?

I apologize for this naive question. Let $\eta$ be an ordinal. Isn't the supremum of $\eta$ just $\eta+1$? If this is true, the supremum is only necessary if you conisder sets of ordinals.
7
votes
3answers
912 views

How is $\epsilon_0$ countable?

In Wikipedia, it says that any epsilon number with the index that is countable is countable. How is it? Out of all those numbers, I especially want to know why $\epsilon_0$ is countable. Thanks.
5
votes
2answers
99 views

Can we implement $\omega^{CK}_1$ using $\omega^{CK}_1+1$ as an oracle?

Let $\omega^{CK}_1$ denote the least non-recursive ordinal. Suppose we have an unknown well-ordering of $\mathbb{N}$ of the order type $\omega^{CK}_1+1$ as an oracle. Is it possible to write an ...
8
votes
4answers
342 views

Is there an element with no fixed point and of infinite order in $\operatorname{Sym}(X)$ for $X$ infinite?

Let $X$ be an infinite set. Let $\operatorname {Sym}(X)$ denote the group of all bijections from $X$ onto itself. I have been thinking about the existence of elements of infinite order in this group. ...
3
votes
2answers
304 views

Proof of two facts on the von Neumann hierarchy by transfinite induction

I have two problems about ordinal numbers. Set $V_0 = \emptyset$ and if we have define $V_a$ then $V_{a+1}= \mathcal{P}(V_a)$ if $a$ is successor and let $ V_{\beta}= \sup\{ V_a ,\quad a<\beta ...
2
votes
3answers
446 views

Closed Intervals of $\omega_1$ Are Compact

Consider the space $\omega_1$ (the first uncountable ordinal) together with the order topology (a convenient base for which is $\{(\alpha, \beta] \mid \alpha, \beta \in \omega_1\} \cup \{0\}$ ). I ...
0
votes
3answers
349 views

union of ordinal number

Which of the following holds: $ \bigcup x \in x ,\quad \bigcup x = x ,\quad x\in \bigcup x $ ,when: (a) $x$ is a set, (b) $x$ is an ordinal. Can someone help me? Thank you in advance!
2
votes
2answers
405 views

Injection from the set of countable ordinals $\Omega$ into $\mathbb{R}$

I'm reading through this and I'd like to define an injective function from the set of countable ordinals $\Omega$ into $\mathbb{R}$ using transfinite induction (or maybe transfinite recursion?). ...
2
votes
1answer
232 views

a question on transfinite recursion on $\mathbf{ON}$

I want to know what's purpose for Kunen to build the theorem I.9.3 on transfinite recursion on $\mathbf{ON}$. Any help will be appreciated. Theorem I.9.3 says: if $F: V \to V$, then there is a unique ...
11
votes
4answers
779 views

I want to know why $\omega \neq \omega+1$.

In Kunen's book, Set Theory,chapter I.7, he said: $1+\omega=\omega \neq \omega+1$. I want to know why $\omega \neq \omega+1$.
1
vote
2answers
139 views

LIM is cofinal in ON

I'd like to show that the class of all limit ordinals, LIM, is cofinal in the class of all ordinals ON. I thought that I could do this by contradiction and assume that it wasn't cofinal in ON. Then ...
2
votes
1answer
269 views

Definition of ordinal

I am confused with the definition of ordinals: "A set $a$ is an ordinal if it is transitive and totally ordered by $\in$." Then an ordinal is a set but we also know that all elements of ordinals are ...
0
votes
1answer
211 views

Formula defining the first transfinite ordinal $\omega$

Just as an exercise in formula manipulation, I tried to find the simplest formula $\phi(x)$ with one free variable $x$ in the language of ZFC that defines the first transfinite ordinal $\omega$ (i.e. ...
5
votes
1answer
338 views

The “canonical” representative of an order type

An ordinal (in von Neumann sense) can be thought as the "canonical" representative of the order type (i.e. of the class of all order-isomorphic sets) of a well-ordered set. This is very convenient, ...
3
votes
1answer
182 views

Showing a $\alpha\times\beta$ is well-ordered for $\alpha,\beta$ ordinals

I am concerned with showing the least element existence in every subset of $\alpha\times\beta$. Below is my attempt. My teacher has used similar argument to show $\mathbb{N}\times\mathbb{N}$ is a ...
1
vote
1answer
352 views

Proof related to an ordinal less than or equal to another ordinal?

Can someone please comment on my solution? I wish to know if my solution is right and every step is well-justified. I will state all propositions that I use in my solution and refer to them later in ...
2
votes
1answer
118 views

Continuous functions on $\omega_1$

Let $\lambda<\omega_1$ be a limit ordinal and consider the set $X=\{\alpha+1\colon \alpha<\lambda\mbox{ is a limit ordinal}\}$ Is the characteristic function $\chi_X$ continuous on $\omega_1$ ...
5
votes
2answers
317 views

Can $\omega_1$ be represented as union of an uncountable collection of cofinal, pairwise disjoint subsets?

Can $\omega_1$ (the first uncountable ordinal) be represented as union of an uncountable collection of cofinal, pairwise disjoint subsets?
1
vote
1answer
137 views

Cardinality of set of normal functions $f \colon \omega_{\alpha} \to \omega_{\alpha}$

What is the cardinality of the set of all normal functions $f \colon \omega_{\alpha} \to \omega_{\alpha}$, where $\omega_{\alpha}$ is the initial ordinal of $\aleph_{\alpha}$?
6
votes
2answers
186 views

Least non-arithmetical ordinal

As I understand, there exists the least ordinal $\alpha$ such that there is no well-ordering of $\mathbb{N}$ which is both order isomorphic to $\alpha$ and is an arithmetical set. Is there a ...
4
votes
1answer
174 views

Products and coproducts in the simplex category

I wonder if the category $\Delta$ of finite totally ordered sets and monotone functions has binary products and coproducts. In particular, is the ordinal sum a categorical coproduct and/or is there ...
0
votes
2answers
467 views

How far do known ordinal notations span?

What is the largest known ordinal number $\alpha$ such that a uniform notation scheme has been developed for all ordinals up to $\alpha$ (there should be no "gaps" in what ordinals are representable), ...
2
votes
2answers
1k views

A proper formal definition of the Ordinal numbers?

Maybe I'm just being a bit stupid with where I'm searching here, but I can't actually find a proper definition of these things wherever I look, books or online. The definition I find most is that it's ...
9
votes
1answer
596 views

Surreal numbers without the axiom of infinity

Let $ZF^\times$ denote the set of axioms of Zermelo-Fraenkel set theory without the axiom of infinity. The set $V_\omega$ of all hereditarily finite sets is a model of $ZF^\times$, and ...
7
votes
2answers
211 views

Complexity of the set of computable ordinals

According to http://en.wikipedia.org/wiki/Analytical_hierarchy The set of all natural numbers which are indices of computable ordinals is a $\Pi^1_1$ set which is not $\Sigma^1_1$. However, "the ...
2
votes
1answer
203 views

A number system

Can we have a number system $S$ of cardinality continuum such that for every $x \in S$, there is a unique $y \in S$, such that for all $z>x$ in S, $x<y\le z$ holds?
9
votes
3answers
788 views

Uncountability of countable ordinals

According to Wikipedia, there are uncountably many countable ordinals. What is the easiest way to see this? If I construct ordinals in the standard way, $$1,\ 2,\ \ldots,\ \omega,\ \omega +1,\ \omega ...
1
vote
1answer
225 views

Mechanical definition of ordinals

It seems that one can construct ordinals from bottom up by successively introducing a new symbol each time a limit is taken: $$1,\ 2,\ \ldots,\ \omega,\ \omega +1,\ \omega +2,\ \ldots,\ \omega\cdot ...
10
votes
3answers
1k views

Is the class of cardinals totally ordered?

In a Wikipedia article http://en.wikipedia.org/wiki/Aleph_number#Aleph-one I encountered the following sentence: "If the axiom of choice (AC) is used, it can be proved that the class of cardinal ...
1
vote
1answer
230 views

About a ordinal-based definition of fast-growing functions

I try to understand the Löb-Wainer-hierarchy and one definition just doesn't open. I hope someone could clarify this to me. A fundamental sequence to limit ordinal $\alpha$ is $\omega$-sequence ...
2
votes
3answers
450 views

What is the ordinal number for the set of binary strings ordered lexicographically?

I was thinking about a CS problem the other day and somehow got on the topic of the order type of the set of all (finite) binary strings ordered lexicographically. I found that I wasn't able to ...
7
votes
4answers
358 views

Supremum and ordinals

Question: Show that sup{$ \xi \dotplus 1 : \xi \in A $} is the least ordinal that is greater than each element of $A$. I tried to get a better feel for this by letting $A=3=${$0,1,2$}. Then I know ...
2
votes
1answer
142 views

Simplify $2^{\omega_1}$

I was given the following task in one exam. The task was as follows: $$\text{Simplify } 2^{\omega_1} \text{ using the following lemma: }1 < \alpha, \beta < \gamma \Rightarrow \alpha^\beta < ...
6
votes
2answers
610 views

Derived sets and ordinals

Given a set $P$ of real numbers, its derived set is the set of all accumulation points - $a\in P^\prime$ if every open set containing $a$ also contains an infinite number of points from $P$ ...
7
votes
4answers
723 views

Simple (even toy) examples for uses of Ordinals?

I want to describe Ordinals using as much low-level mathematics as possible, but I need examples in order to explain the general idea. I want to show how certain mathematical objects are constructed ...
7
votes
1answer
171 views

Sequence of surjections imply choice

I am reading a paper where a side remark said that if a sequence $\langle g_\beta\colon\omega\to\beta\mid\beta<\omega_1\rangle$ is a sequence of surjections then $\omega_1$ is regular. I have ...