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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4answers
197 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
179 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
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3answers
236 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 ...
1
vote
3answers
155 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
277 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?).
...
10
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4answers
599 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
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2answers
118 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
187 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
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1answer
151 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. ...
3
votes
1answer
175 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
149 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
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1answer
217 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
101 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 $\omega_1$ ...
3
votes
2answers
192 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
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1answer
108 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
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2answers
143 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 ...
2
votes
1answer
114 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
267 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), ...
1
vote
2answers
312 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 ...
8
votes
1answer
424 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
155 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
155 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?
7
votes
2answers
340 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,
$$1,\ 2,\ \ldots,\ \omega,\ \omega +1,\ \omega ...
1
vote
1answer
167 views
Mechanical definition of ordinals
It seems that one can construct ordinals from bottom up by successively introducing a new symbol each time a limit is taken:
$$1,\ 2,\ \ldots,\ \omega,\ \omega +1,\ \omega +2,\ \ldots,\ \omega\cdot ...
7
votes
3answers
519 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
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1answer
178 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
252 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
214 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
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1answer
120 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
260 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
3answers
459 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 ...
5
votes
1answer
139 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 ...
12
votes
1answer
228 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 ...
10
votes
1answer
442 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
120 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
272 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
156 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
97 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 ...
5
votes
2answers
249 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" ...
7
votes
1answer
279 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
530 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
104 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 ...
12
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 ...
5
votes
2answers
722 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 $n$ ...
7
votes
4answers
946 views
Simple example of uncountable ordinal
Can you make a simple example of an uncountable ordinal? With simple I mean that it is easy to prove that the ordinal is uncountable. I know that the set of all the countable ordinals is an ...
5
votes
1answer
468 views
Uncountable ordinals without power set axiom
Assume $M$ is a set, in which all axioms of $ZF - P + (V=L)$ hold. Does then $M$ believe that there exists an uncountable ordinal? I mean, why should the class of all countable ordinal numbers be a ...
