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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Embedding $\omega_1$ in the hyperreals

I've been trying to show that there exists a subset of the hyperreals which has order type $\omega_1$, that is, that there exists a subset $A \subseteq {}^*\mathbb{R}$ for which there is an order ...
3
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0answers
20 views

Ordinal exponentiation identity with natural sum of exponents

This is related to a previous question on How to think about ordinal exponentiation? One possible definition for the natural product $\alpha\otimes\beta$ of ordinals is based on Cantor Normal Forms ...
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26 views

Least ordinal $\alpha$ that $\omega^\alpha > \beta$ is a successor ordinal

Let $\beta$ be an ordinal. The least ordinal $\alpha$ that $\omega^\alpha > \beta$ is a successor ordinal. This is true, but I'm not sure why. Can someone give me a hint?
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1answer
60 views

How to deal with probability problems in proper class sample spaces?

Here my focus is mainly on $Ord$ and questions such as: An ordinal is chosen by random. What is the probability of the event that it is a cardinal number? A coin is tossed proper class many ...
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4answers
119 views

Is $0^\omega=1$?

According to a definition of ordinal exponentiation defined in Kunen's Set Theory: An Introduction to Independence Proofs (pp. 26), we define $$\begin{align} \alpha^0&=1\\ ...
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1answer
27 views

to show that there is no injection from a finite successor of finite ordinal to itself

im trying to prove this proposition let $\alpha \in \omega$ so i want to show that there is no injection (not necessarily keeps order) from $\alpha +1$ to $\alpha$ without using cardinality or the ...
6
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1answer
39 views

Why are all von Neumann ordinals contained in each other?

I'm a high school student and I'm giving a lecture in my high school math class on ordinal numbers, and I would like to prove that the von Neumann ordinals are well-ordered by set membership. The ...
3
votes
1answer
381 views

How to prove that Cantor's normal form can produce all ordinal numbers

How do we prove Cantor's normal form can produce all ordinal numbers? Also, it's a bit difficult to picture $\omega_1$ as a format in Cantor's normal form. Can anyone show how to do this?
6
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1answer
79 views

Regarding functions on $\omega_1$

I've been trying to prove a property that apparently all functions $g: \omega_1 \rightarrow \omega_1$ have, where $\omega_1$ is the least uncountable ordinal. For $\alpha \in \omega_1$, define ...
17
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4answers
556 views

Intuition for $\omega^\omega$

I'm trying to understand the ordinal number $\omega^\omega$ and I'm having a hard time. I think I understand what $\omega^2$ is. It's what I would get if I took countably many copies of $\omega$ and ...
5
votes
2answers
299 views

Equivalent characterizations of ordinals of the form $\omega^\delta$

Let $\alpha$ be a limit ordinal. Show the following are equivalent: $\forall \beta, \gamma<\alpha (\beta+\gamma<\alpha)$ $\forall \beta<\alpha(\beta+\alpha=\alpha)$ $\forall X\subset ...
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1answer
43 views

Are well-orders of the same recursive length recursively isomorphic?

If the ordinal length of $A$ and $B$ is the same recursive ordinal, does it follow that there is a recursive one-one order-preserving correspondence between $A$ and $B$?
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2answers
410 views

Cardinal Arithmetic versus Ordinal Arithmetic

I am reading Philosophy, not Set Theory, so please excuse the naivety of my question. My question concerns the wildly different character of ordinal arithmetic versus cardinal arithmetic. The ...
0
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1answer
28 views

Example of a well-ordered set with a specific order-type

An example of a set with order type $\omega^2$ is $\mathbb{N}\times\mathbb{N}$ with a lexicographic order. An example of a set with order type $\omega^3$ is ...
0
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1answer
66 views

Ordering ordinals by size [closed]

Well define $\omega,\omega_1, \omega_2$ to be the first three infinite ordinals. Order them according to their size: $2\cdot\omega_1+\omega\cdot3+3,$ $\omega\cdot3+\omega_1+3,$ ...
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3answers
225 views

Other ways of proving that the set of all countable ordinals is uncountable

I know that the standard way of proving that the set of all countable ordinals is uncountable is by stating that if the set is countable, then it incurs Burali-Forti paradox. Is there other ways of ...
2
votes
1answer
62 views

(If exists) a set of all ordinals that set is an ordinal?

In set theory (ZF) an ordinal is a transitive set of transitive sets. Thus (if exists) a set of all ordinals gives a contradiction therefore there is no set of all ordinals. But what is wrong with ...
3
votes
3answers
63 views

Proving that $ω_1$ is locally compact

I'm trying to show that $ω_1$ is locally compact, but when doing so, I need to show something else, which got me a bit stuck on. I'm taking a $\alpha\in ω_1$, so $\{\alpha\}$ is an open set. Since ...
3
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1answer
53 views

Is $\omega_1$ metrizable?

Following Urysohn's metrization theorem, I would like to prove or disprove that $\omega_1$ is metrizable. I know it is hausdorff, but I'm not sure whether or not it is second countable, and I'm at ...
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0answers
22 views

Subsets of $ \mathbb Q $ of order type $ \omega^{\alpha}$ for each countable ordinal $\alpha $.

My introductory text in Set Theory (Stillwell) includes an exercise (6.3.1) asking for an explicit example of a subset of $ \mathbb Q $ or order type $ \omega^2 $. This seems straight forward enough. ...
4
votes
3answers
141 views

An easy to understand definition of $\omega_1$?

I have two things I'm not sure in 100% about them. The first, is $\omega_1$. I have a little "feeling" of it, but if I'll be asked to define it - I don't know where to begin from. Perhaps it is ...
1
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1answer
63 views

If $X$ is a subset of $\omega_{\alpha}$ such that $|X| < \aleph_{\alpha}$, then $|\omega_{\alpha} - X| = \aleph_{\alpha}$

If $\alpha=0$, then $\omega_0=\mathbb{N}$ and $\aleph_0=$ countable. So $|\omega_{\alpha} - X| = \aleph_{\alpha}$ becomes $|\mathbb{N}-X|=|\mathbb{N}|$ which is true (the function $f:\mathbb{N} ...
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0answers
43 views

Ordinals as Trees

I'm trying to understand countable ordinals and their tree representation. I understand that $\omega$ is the first "non branching tree" of infinite height. I also understand that the exponent of ...
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0answers
28 views

Comprehending ordinals: from $\omega^\omega$ through $\omega^{\omega^2}$ to $\varepsilon_0$

I am currently trying to comprehend ordinal numbers by finding an order on some countable set (like natural numbers or tuples of natural numbers) that is isomorphic to some ordinal. For an instance, ...
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0answers
39 views

Chinese remainder theorem for infinite equations & equations of infinities

The Chinese remainder theorem is a very well-known theorem of arithmetic and algebra. Inspired by the following paper in model theory, Li Rong Yu, Li Bo Luo - The Generalization of the Chinese ...
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0answers
59 views

Prove that ordinal multiplication is left distributive

Suppose $\alpha, \beta$ and $\gamma$ are ordinals. Prove the distributive law $\alpha \cdot ( \beta + \gamma) = \alpha \cdot \beta + \alpha \cdot \gamma$. The following is my proof: Proof: We use ...
2
votes
1answer
78 views

A few questions on $\omega_1$

I'm trying to understand several things regarding $\omega_1$ and trying to get a better feeling of it. First question is - is $\omega_1$ a connected space? I think it isn't, but not really sure how ...
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2answers
46 views

What is the smallest infinite ordinal that is not order isomorphic to a reordering of the natural numbers?

I've been working on a particular set theory problem for a while and essentially I've hit a roadblock because of this question. I just need to know what this ordinal is, because I have a sneaking ...
1
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3answers
59 views

the smallest transfinite ordinal number

My guess is that $\omega$ is the smallest transfinite ordinal number. To prove this, let $\beta$ be any transfinite ordinal number and let $\operatorname{ord}(B, \le)= \beta$. And I need to show that ...
2
votes
1answer
41 views

Cardinality: is it true that $|X^\mathcal{inj}| \leq 2^{|X|}$?

Definition. Whenever $X$ is a set, write $X^\mathcal{inj}$ for the collection of all injections $f$ such that: $f$ has codomain $X$ There exists an ordinal $\alpha$ such that $f$ has domain ...
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3answers
342 views

How to prove that the set of all countable ordinals, $\omega_1$, is uncountable / has the cardinality $\aleph_1$?

The only reasoning I've seen given for this is that it's uncountable because it can't include itself an element. I'm a little unconvinced and was looking for a more proper formal proof demonstrating ...
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1answer
18 views

In what sense $\alpha \times \alpha$ is the initial segment $(0,\alpha)$ in $Ord \times Ord$?

This is from Jech's book on set theory: "We define a well ordering of the class $Ord \times Ord$ of ordinal pairs. Under this well ordering, each $\alpha \times \alpha$ is an initial segment of ...
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0answers
37 views

Ordinal arithmetic and functions

I have two function $G$ and $F$ defined on ordinals and I know that $$G(\alpha +\omega )\subseteq F(\gamma +\alpha+\omega)$$ when $G(\alpha)\subseteq F(\gamma)$ and $\alpha$ is a limit ordinal. I ...
0
votes
1answer
44 views

Definition of continuity of ordinal function

In the book Introduction to Set Theory' by Hrbacek and Jech, chapter $6$ Ordinal Numbers, section $6$ Normal Form, I don't understand the definition of continuity of ordinal numbers. Ordinal ...
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0answers
47 views

show that ordinal multiplication is associative

Prove that ordinal multiplication $\alpha \cdot (\beta \cdot \gamma) = (\alpha \cdot \beta) \cdot \gamma$ is associative by using the following facts: $$\beta \cdot 0=0$$ $$\beta \cdot (\alpha ...
3
votes
2answers
71 views

Embedding $\omega_1$ into the direct sum/product of $\Bbb R$'s and $\Bbb N$'s

I'm trying to find a way to visualize the first uncountable ordinal $\omega_1$. This is rather difficult, as the visualization tactic that I often use for countable ordinals - namely, the "matchstick" ...
0
votes
0answers
29 views

'Popular mathematics' resource about infinite ordinals

The 'popular mathematics' literature (think Martin Gardner, William Dunham, Hofstadter, and the like) abounds with material on the mathematics of infinite cardinals, starting - and quite often ending ...
0
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1answer
20 views

For every ordinal $\alpha$, there is a cardinal number greater then $\alpha$.

I am trying to prove that for every ordinal $\alpha$, there is a cardinal number greater then $\alpha$. Proof: $\alpha$ is a (well ordered) set. Take the set $P(\alpha)$. Assume that we know that for ...
1
vote
1answer
20 views

A limit ordinal $\gamma$ is indecomposable if and only if $\alpha + \gamma = \gamma$ if and only if $\gamma = \omega^{\alpha}$ for some $\alpha$. [duplicate]

I am trying to prove ex 2.13 from Jech's book on set theory: Ex 2.13: A limit ordinal $\gamma$ is indecomposable if and only if $\alpha + \gamma = \gamma$ if and only if $\gamma = \omega^{\alpha}$ ...
3
votes
0answers
40 views

If a linearly ordered set $L$ has the property that every order-preserving injection $L \rightarrow L$ is expansive, is $L$ necessarily well-ordered?

Given a poset $P$, call a function $f : P \rightarrow P$ expansive iff $f(x) \geq x,$ for all $x \in P$. Now suppose a linearly ordered set $L$ has the property that every order-preserving injection ...
1
vote
1answer
39 views

Introduction to Set Theory(Ordinal Numbers)

Lemma: If $\alpha$ is an ordinal number, then $\alpha \not \in \alpha$ Proof: If $\alpha \in \alpha$, then the linearly ordered set $(\alpha, \in_{\alpha})$ has an element $x=\alpha$ such that $x \in ...
0
votes
1answer
44 views

Question regarding addition of order types

I have been reading through Herbert Endertons introductory book on set theory as I have stumbled upon a claim that baffled me through the day. As anyone I need sleep so I ask here for help. Namely ...
1
vote
4answers
131 views

Are $1+ω$ and $ω+1$ isomorphic?

In set theory, $1+ω$ is defined as the ordinal number of ordinal sum of sets $\{a\}$ and ℕ. Also $ω+1$ is defined as the ordinal number of ordinal sum of sets $\Bbb N$ and $\{a\}$. You know what the ...
4
votes
2answers
327 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 ...
0
votes
1answer
31 views

Ordinal multiplication property: $\alpha<\beta$ implies $\alpha\gamma\le\beta\gamma$

I'm having trouble proving the following two ordinal multiplication properties. If $\alpha, \beta$, and $\gamma$ are such that $\alpha \lt \beta$ and $\gamma \gt 0$, then $\alpha\gamma \le ...
2
votes
2answers
56 views

Ordinal numbers addition property: $b<c$ implies $b+a \le c+a$

I'm having trouble proving the following property of ordinal numbers. If $a, b, c$ are ordinal numbers such that $b \lt c$, then $b+a \le c+a$. I first started by assuming $g$ as an order ...
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2answers
49 views

$\kappa\cdot\kappa= \kappa,$ for infinite cardinals

I am trying to understand the proof that uses a maximal-lexicographic ordering. For an infinite ordinal $\kappa,$ the canonical well-ordering of $\kappa \times \kappa,$ denoted by $<_{cw}$ is ...
1
vote
1answer
62 views

Is it possible to iterate a function transfinite times?

Let $f:A\rightarrow A$ be a function. We can simply define $f\circ f$, $f\circ f\circ f$, etc., for each given natural number inductively. $f^{(0)}=id_A$ $\forall n\in\omega \qquad f^{(n+1)}=f\circ ...
14
votes
2answers
232 views

The preorder of countable order types

Consider the set $\mathcal{O}$ of order types corresponding to all posets of cardinality at most $\aleph_0$. The set $\mathcal{O}$ is a preorder under embeddability of its elements (note that some ...
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2answers
36 views

How does cf(cf $\alpha$) = cf $\alpha$?

Assume $\alpha > 0$ is a limit ordinal, and cf $\alpha=$ the least ordinal $\beta$ such that there is an increasing $\beta$-sequence $\langle \alpha_\xi \, \colon\, \, \xi < \beta\rangle$ that ...