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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93 views

Basic examples of ordinals

I'm reading a book on stochastic games which apparently needs ordinals somewhere. This is the first time I meet a concept, and although the definition is clear to me, the lack of practice makes it ...
0
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1answer
58 views

Definition of ordinal exponentiation

I found that the usual ordinal exponentiation $\alpha^{\beta}$ is the set of functions from $\beta$ to $\alpha$ with finite support, ordered by antilexicographic order. (least significant position ...
3
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1answer
70 views

Base change and ordinals

Problem. Define the operation base change from $k$ to $m$: to make the operation for natural number $n$ we should write $n$ in the base-$k$ numeral system and read this in the base-$m$ numeral ...
5
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0answers
175 views

Largest provably existing ordinal in ZF without power set

If I remove the power set axiom from ZF, but retain the axiom of infinity, what will be the largest ordinal that can still be proven to exist. Or, if no largest such ordinal exist, what is the limit ...
4
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3answers
226 views

Countable ordinals are embeddable in the rationals $\Bbb Q$ — proofs and their use of AC

Yesterday, Asaf Karagila's answer to my question sparked an extensive discussion on ways of proving that all countable ordinals are embeddable in $\Bbb Q$, and whether particular solutions to this use ...
5
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1answer
60 views

Equivalence between these definitions of ordinal numbers

Von Neumann defines an ordinal $\alpha$ as a transitive set whose elements are well-ordered with respect to the membership relation $\in$. Meanwhile, in Naive set Theory, Halmos defines an ordinal ...
8
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1answer
136 views

Do there exist totally ordered sets with the 'distinct order type' property that are not well-ordered?

Define that the order type of an element $x$ in a totally ordered set $X$ is the order type of $\{w \in X\mid w < x\}$. Under this definition, distinct elements of a well-ordered set have distinct ...
4
votes
1answer
208 views

Ordinal arithmetic $3+\omega^2 = \omega^2$

Which one of the following equalities is false: a) $2\cdot \omega = 3\cdot\omega$ b) $3+\omega+\omega^2 = \omega+3+\omega^2$ c) $\omega^2 + 3 = 3+\omega^2$ d) $12\cdot(5+\omega)=60 \cdot\omega$ I ...
10
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3answers
189 views

How many epsilon numbers $<\omega_1$ are there?

An epsilon number is an ordinal $\epsilon$ such that $\epsilon=\omega^\epsilon.$ What is the cardinality of the set of all epsilon numbers less than $\omega_1$? I'm asking this because of a proof ...
5
votes
1answer
235 views

What set theory axioms do I need to believe in uncountable ordinals?

Math people: The title is the question. I am convinced that uncountable sets exist, thanks to Cantor's diagonal proof. It is not intuitively clear to me that uncountable ordinals or cardinals should ...
2
votes
1answer
117 views

Prove $\omega^{\omega_1}=\omega_1$ and $2^{\omega_1}=\omega_1$

Prove that $\omega^{\omega_1}=\omega_1$ and $2^{\omega_1}=\omega_1$. I also found an exercise asking to compute the ordinal number $\omega_1^{\omega}$, but I do not even understand what I am supposed ...
5
votes
3answers
143 views

How to show that $(\Bbb Q, <)$ contains an order isomorphic copy of $\epsilon_0$ which is the least ordinal satisfies $\omega^\epsilon = \epsilon$?

We define $\epsilon_0 = \sup\{\omega, \omega^\omega,\omega^{\omega^\omega},\omega^{\omega^{\omega^\omega}},\ldots\}$. How to find a subset of $ \Bbb Q$, $(A, <_{\Bbb Q})$ that is isomorphic to ...
5
votes
1answer
102 views

$\mathcal{P}(\omega \times \omega)$ contains a copy of every countable ordinal

I am trying to understand this proof of the existence of an uncountable ordinal. I don't see why $\mathcal{P}(\omega \times \omega)$ contains a copy of every countable ordinal as it is said. For ...
2
votes
1answer
63 views

Simple ordinal number question

I got an excercise to add some ordinals, and one of given ordinals was: $$\omega\cdot4\omega^2+1.$$ I know that $4\omega = \omega$ and that multiplication is associative, so I think this simply equals ...
4
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4answers
230 views

Does every ordinal have cardinality no greater than $\aleph_\mathbb{0}$?

My notes say that the ordinals $\omega + 1, \omega + 2, ... , 2 \omega, ... , 3 \omega, ... \omega^2, ... $ are all countable, and hence have cardinality equal to $\omega = \aleph_\mathbb{0}$. So I ...
5
votes
1answer
91 views

Does the definition of countable ordinals require the power set axiom?

I am trying to understand the consequences of the different axioms of ZFC. In particular, I was trying to understand what you get on ZFC-power set (ZFC minus the power set axiom). If you have any ...
3
votes
2answers
255 views

Why does every countable limit ordinal have cofinality $\omega$?

According to Wikipedia, if $\alpha$ is a countable limit ordinal, then $\mathrm{cf}(\alpha)=\omega$. It is intuitively clear to me that it should be so. Certainly the cofinality of such an ordinal ...
0
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2answers
170 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 ...
7
votes
1answer
153 views

Do we get predicative ordinals above $\Gamma_0$ if we use hyperexponentiation?

I am trying to understand the Veblen hierarchy but I still find it confusing. The Feferman–Schütte ordinal, $\Gamma_0$, can be described as the set of all ordinals that can be written as finite ...
7
votes
1answer
137 views

When the ordinal sum equals the Hessenberg (“natural”) sum

Let $\alpha_1 \geq \ldots \geq \alpha_n$ be ordinal numbers. I am interested in necessary and sufficient conditions for the ordinal sum $\alpha_1 + \ldots + \alpha_n$ to be equal to the Hessenberg ...
2
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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 ...
4
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0answers
51 views

Nimber of selective compound games

Background/Definitions. Let $\alpha,\beta$ ordinal numbers. The Hessenberg sum $\alpha \# \beta$ is defined recursively as the smallest ordinal which is $>\alpha' \# \beta$ and $> \alpha \# ...
6
votes
2answers
190 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 ...
3
votes
1answer
164 views

Diagonal intersection of club sets

Let $C_\alpha\subset\omega_1$ be a club set for every $\alpha<\omega_1$. Show that the diagonal intersection of all the $C_\alpha$'s, that is $\{\alpha<\omega_1:\forall\beta<\alpha,\alpha \in ...
2
votes
1answer
105 views

Transfinite sequence

I have only little knowledge concerning ordinals and transfinite induction, but I encountered a definition in a paper where a countably infinite set $H$ (the set consists of formulae, but this does ...
2
votes
2answers
110 views

Another question about Cohen's article

This is another question related to Cohen's Discovery of Forcing: Consider the following passage from p. 1091: Suppose $M$ were a countable model. Up until now we have not discussed the role ...
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 $\omega_1$?
19
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1answer
270 views

What are some good open problems about countable ordinals?

After reading some books about ordinals I had an impressions that area below $\omega_1$ is thoroughly studied and there is not much new research can be done in it. I hope my impression was wrong. ...
13
votes
3answers
356 views

How many positive numbers need to be added together to ensure that the sum is infinite?

The question in the title is naively stated, so let be make it more precise: Let $\sum_{n\in\alpha}a_n$ be an ordinal-indexed sequence of real numbers such that $a_n>0$ for each $n\in\alpha$, where ...
2
votes
2answers
125 views

Ordinals Addition Property

I’m sill trying to figure out how to prove that the following property of ordinals holds Let $x, y, z \in$ On where On is the class of ordinals. If $x < y$ then $\forall z$ we have that ...
3
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0answers
45 views

Soft Question: Scientific applications of ordinal arithmetic?

Are there any known scientific applications of ordinal arithmetic -- either direct applications or application of results in other areas that depend even indirectly on results from the study of ...
16
votes
1answer
215 views

When does ordinal addition/multiplication commute?

I'm looking at basic ordinal arithmetic at the moment, and I am aware that in general, $\alpha+\beta\neq\beta+\alpha$ and $\alpha.\beta\neq\beta.\alpha$ for ordinals $\alpha,\beta$. My question is: ...
1
vote
0answers
36 views

Is the space $X$ in the class dual to the spaces with the Souslin property?

Recall that $X$ is in the class dual to the spaces with the Souslin property: For any neighbourhood assignment $\{O_x: x\in X\}$, there is a subspace $Y \subseteq X$ such that $c(Y)=\omega$ and ...
3
votes
2answers
102 views

Omega, well ordering and the cumulative hierarchy $V$.

Hi I need help with a problem of set theory. I'm not sure how to prove that the well ordering on $\omega$ isomorphic to $\omega+\omega$ belongs to the level $V_{\omega+\omega}$ in the hierarchy $V$. ...
1
vote
3answers
89 views

How to prove $X$ is normal?

Let $X=[0,\omega_1]\setminus \{\omega_1\}=[0,\omega_1)$. I known every linear order space is hereditarily normal. However, I would like to know the easier proof by which we can conclude that $X$ is ...
4
votes
2answers
77 views

Bijection between $1+ \alpha$ and $\alpha + 1$

Let $\alpha$ be an ordinal. I want to show that there is a bijection between the ordinals $1 + \alpha$ and $\alpha + 1$. I tried to proceed by transfinite induction on $\alpha$; in case $\alpha$ is a ...
3
votes
1answer
68 views

Are there structures other than orderings that refine cardinality?

Two sets are of the same cardinality if a bijection exists between the two sets. Two sets represent the same ordinal if an order preserving bijection exists between them. The ordinals produce ...
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?
1
vote
1answer
48 views

How to prove $\gamma < \omega_1$?

Let $\gamma=\sup\{\xi_n\}=\{\xi_n<\omega_1: n\in \omega\}$. How to prove $\gamma < \omega_1$? Thanks for your help.
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 $\aleph_\alpha$ ...
6
votes
2answers
277 views

Is the proper class of all ordinals equivalent to the potential infinity of pre-Cantor times?

My understanding is that the class of all ordinals is, by definition a proper class. This in the end is done to avoid a paradox: the collection of all sets would be paradoxical if you allow it to be a ...
3
votes
2answers
95 views

Is the set of countable successor ordinals countable or uncountable?

I know that $\omega_1$={$\alpha$ : $\alpha$ is a countable ordinal} is uncountable but what about the subset of $\omega_1$ of countable successor ordinals?
1
vote
2answers
181 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
3answers
95 views

Limit Ordinal Question

how to prove this result. $\alpha$ is limit ordinal if and only if $\beta<\alpha$ implies $\beta+1<\alpha$ for any $\beta$. I am getting confused because of the definition of limit ...
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 ...
3
votes
3answers
94 views

What are the sets $S_n=\omega-n$ called?

What are the sets $S_n$ where $S_n:=\omega-n$ called? I explain better: if ordinals are defined in this way $0=\varnothing$ $1=\{\varnothing\}=\{0\}$ $2=\{0,1\}$ $n=\{0,1,..,n-1\}$ ...
4
votes
2answers
156 views

Indecomposable limit ordinals

A limit ordinal $\gamma>0$ is said to be indecomposable iff $\nexists\alpha,\beta<\gamma$ such that $\alpha+\beta=\gamma$. In view of this definition, I’m trying to prove the equivalence of the ...
8
votes
2answers
243 views

How many ordinals can we cram into $\mathbb{R}_+$, respecting order?

I've been pondering the following question. How can we measure the amount of "space" above an element $p$ in a partially ordered set $P$? One way would be to try to cram the elements of ...
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 ...