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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163 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
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2answers
362 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
872 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
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2answers
97 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
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4answers
321 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
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2answers
300 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
440 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
342 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
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2answers
394 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
231 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 ...
10
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4answers
772 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 $\omega \neq \omega+1$.
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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
207 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
333 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
347 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
310 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
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1answer
159 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
454 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
587 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
206 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
202 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?
8
votes
3answers
755 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
222 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
228 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
438 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
356 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
141 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
581 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
697 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 ...
13
votes
1answer
375 views

How do you find the smallest of homeomorphic ordinals?

I am trying to get a better feel for the topology of ordinals and just received a great answer to this question where the Cantor-Bendixson rank and degree turn out to be a complete homeomorphism ...
14
votes
3answers
1k views

Countable compact spaces as ordinals

I heard at some point (without seeing a proof) that every countable, compact space $X$ is homeomorphic to a countable successor ordinal with the usual order topology. Is this true? Perhaps someone can ...
6
votes
0answers
204 views

Computable increasing function from $\omega_1^{CK} \to \mathbb{R}$

If $f:\omega_1 \to [0,1]$ such that $f(0)=0$ and such that $f(\alpha)<1$ and $\alpha<\beta$ imply $f(\alpha)<f(\beta)$ then there is a $\gamma$ such that $f(\gamma)=1$. Non-constructive ...
4
votes
1answer
336 views

Equivalent definitions of ordinals?

The first definition of an ordinal number I found was that an ordinal number is the $\in$-image of a well-ordered set $(A,\lt)$. From this definition it was derived that an ordinal is just the set of ...
3
votes
1answer
193 views

If $\kappa<\aleph_\alpha$, then $\kappa\leq\alpha$?

I'm having a hard time following this proof. Here $\aleph(\alpha)$ is the cardinality of $Z(\alpha)$, the set of all ordinals $\gamma$ such that $|\gamma|\leq\alpha$. Also, $\aleph_1=\aleph(\aleph_0)$ ...
3
votes
1answer
111 views

Small question on proof that any ordinal is the index of some initial ordinal

I'm reading a proof of a theorem, but an apparently trivial case has tripped me up. The theorem goes as For any ordinal $\alpha$, there is a initial ordinal $\phi$ such that $i(\phi)=\alpha$. Here ...
7
votes
2answers
335 views

Is $V$ under ZFC really a proper class?

Is $V$, the union of the von Neumann hierarchy, necessarily a proper class? Or is it only a proper class after you assume that it contains every set? (In that case, $V$ can't be a "set of all sets" ...
8
votes
1answer
453 views

Can any infinite ordinal be expressed as the sum of a limit ordinal and a finite ordinal?

I've been browsing through Jech's and Levy's texts on set theory, and the ideas of ordinals come up fairly quickly. The idea of a limit ordinal is introduced, which is an ordinal with no maximum ...
3
votes
1answer
869 views

A transitive set of ordinals is an ordinal

This is Exercise III.2.20 of Bourbaki's Set Theory. (Von Neumann ordinals are actually called "pseudo-ordinals" by Bourbaki, but I simply call them ordinals here) Let $X$ be a transitive set, and ...
5
votes
2answers
115 views

Transfinite sequences of topologies

I am not that familiar with arguments using ordinals so this may be pretty simple. Let $T$ be a topology on a set $X$. For each ordinal $\alpha$ let $T_{\alpha}$ be a topology on $X$ satisfying ...
6
votes
1answer
170 views

Proving $0$ is an ordinal

In Introduction to Set Theory by J. Donald Monk, he defines ordinal as follows. Definition (1): $A$ is an ordinal iff $A$ is $\in$-transitive and each member of $A$ is $\in$-transitive. $A$ is ...
14
votes
1answer
2k views

The cardinality of a countable union of countable sets, without the axiom of choice

One of my homework questions was to prove, from the axioms of ZF only, that a countable union of countable sets does not have cardinality $\aleph_2$. My solution shows that it does not have ...
7
votes
2answers
866 views

Non-existence of a surjection $\aleph_n \to \aleph_{n+1}$, without the axiom of choice

Firstly, let's establish what exactly I mean by these symbols. Let $\omega_0 = \{ 0, 1, 2, \ldots \}$, where $0, 1, 2, \ldots$ are the usual von Neumann representations of the natural numbers. Let $n$ ...