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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445 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), ...
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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
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1answer
580 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
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2answers
205 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 ...
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1answer
201 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?
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739 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 ...
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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 ...
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3answers
978 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 ...
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1answer
226 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
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3answers
427 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
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4answers
354 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
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 < ...
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2answers
568 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
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4answers
681 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
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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
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1answer
372 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
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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
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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
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1answer
333 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
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1answer
192 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
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1answer
110 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 ...
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2answers
331 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" ...
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1answer
435 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
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1answer
863 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 ...
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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
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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
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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
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2answers
861 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 $n$ ...
8
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4answers
1k 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 ...
4
votes
2answers
372 views

Order-isomorphic with a subset iff order-isomorphic with an initial segment

Let $(X, \prec)$ and $(Y, <)$ be two well-ordered sets. I want to show that if $X$ is isomorphic to a subset of Y then $X$ is isomorphic with an initial segment of $Y$. (The other direction is of ...
7
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1answer
715 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 ...