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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145 views

Raising a partial function to the power of an ordinal

Consider a set $X$, and let $f : X \rightarrow X$ denote a partial function. Then for natural $n$, we can define $f^n$ as iterated composition, e.g. $f^2 = f \circ f$. Now suppose that $X$ is also ...
9
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310 views

The structure of countable ordinals

Consider the recursively defined hyperoperation sequence $\circ_i$ $$\begin{array}{rcrclclcl} x& \small{+}&(y\ {\small+}1)&:=&x& &&{\small+}&1\\ x& \boldsymbol{+}&...
8
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147 views

Lebesgue Premeasure via Transfinite Induction

If $I=[a,b)$ we write $|I|=b-a$ for the length of $I$. Given a theorem of Caratheodory, the tricky part in showing the existence of Lebesgue measure is this: Lemma If $[0,1)$ is the disjoint union of ...
7
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147 views

What well-orders are definable over $V$?

Let $V$ be a (the) universe of sets, and $On^V$ denote the ordinals of $V$. It is well known that there are formulae that seem to define orderings `longer than' $On^V$. For example: $\alpha < \...
7
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139 views

Order-preserving map of regressive functions on $\omega_1$

I posted the following question a year ago on MO. It did receive some attention, but the answer there remains incomplete (as discussed in some detail in its comments). Meanwhile I got the impression ...
6
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249 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 ...
6
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212 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 proof:...
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93 views

Is it possible to formalize all mathematics in terms of ordinals only?

Our experience shows that all finitary mathematical objects could be encoded using the natural numbers, and all operations on those objects could be expressed in terms of a few basic operations on ...
4
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216 views

Visualizations of ordinal numbers

I find this picture of the ordinal numbers up to $\omega^\omega$ rather hard to grasp: I wonder if the following might be a more compelling way to visualize ordinal numbers up to $\omega^\omega$: ...
4
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71 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 \# \...
4
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66 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 ...
4
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123 views

Transfinite induction and valuation rank

In Engler's valued fields, exercise 3.5.2 goes as follows Construct valuations on $\Bbb{C}$ of rank $\kappa$ for every cardinal $\kappa \leq 2^{\aleph_0}$. The idea behind this (for any ...
3
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54 views

The supremum of any set of cardinals (considered as a set of ordinals) is again a cardinal.

An ordinal $\alpha$ is a cardinal iff no $\xi < \alpha$ is equivalent to $\alpha$. Now, let $A$ be any set of cardinals and $\sup(A)=\alpha$, then for $\xi < \alpha$ there is a $\beta \in A$ ...
3
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86 views

Height of an ordered field

I'm studying ordered fields, and a specific notion regarding ordered fields that I will denote here by their "height". If $k$ is an ordered field, and $\alpha$ is a non-empty ordinal, a ruler of ...
3
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55 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 $...
3
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312 views

How to derive Church-Kleene ordinal

Crossing-out: (How does one prove the existence of Church-Kleene ordinal? Also, why is it labeled as $\omega_1^{CK}$? And why is it first ordinal not hyperarithmetical, and is the first admissible ...
2
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48 views

A question on ordinal arithmetic.

I have to order these two ordinals and I was just wondering if I have done it correctly. $\omega^\omega + \omega^3$ and $\omega + \omega^3 +\omega^\omega$ I have worked out that $\omega + \omega^3 +...
2
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60 views

Trichotomy of Ordinals. Is $K$ a set?

We want to prove the "Trichotomy of Ordinals": Definiton: An ordinal is a transitive set with elements that are all transitive. Definiton: $\alpha$ and $\beta$ ordinals are comparable if one of the ...
2
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45 views

Is $(\omega \times \omega)^{\omega}\cong \omega \times \omega \times… \cong \omega^{\omega}$? Where “$\cong$” means homeomorphic.

I'm interested in the circumstances for when we can conclude that two ordinal spaces are homeomorphic by an examination of their written form. Specifically, I'm taking an ordinal space, say $(\omega^...
2
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0answers
27 views

Completion of surreal subfields

Let $\kappa$ be a regular uncountable ordinal. Let $No(\kappa)$ denote the field of surreal numbers of birthdate $ < \kappa$. In Fields of surreal numbers and exponentiation (2000), P. Ehrlich and ...
2
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32 views

show that the von Neumann universes $R_{\omega+ \alpha}$ have cardinality $\beth_\alpha$.

Trying to show that the von Neumann universes $R_{\omega+ \alpha}$ have cardinality $\beth_\alpha$. Here is my attempt: Proof by induction on $\alpha$. For $\alpha = 0$, $|R_{\omega+ 0}| = |R_{\...
2
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71 views

Compact, sequential spaces

A compact, Hausdorff space $X$ is sequential if each for each $A\subset X$ and $x\in \overline{A}$, there exists a countable set $A_0\subset A$ such that $x\in \overline{A}_0$. I am asked to show ...
2
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164 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. ...
2
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109 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 ...
2
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47 views

Prove: Suppose $\alpha > 1$ and $\beta, \gamma$ are ordinals with $\beta < \gamma$. Then $\alpha^\beta < \alpha^\gamma$.

In this question, $\alpha$, $\beta$ and $\gamma$ are ordinals. I want to prove this by transfinite induction on $\gamma$, which typically has two or three cases. I'm considering three cases: the base ...
2
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46 views

Find ordinal of a strange ordered set

I was asked the following question: for every $x\in \mathbb R$ we define the set $Q(x)=\{q \in \mathbb Q|q\leq x\}$, the set of all rational numbers less or equal to $x$. Let $M=\{Q(x)|x\in \mathbb ...
2
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29 views

what is the third term of the $\omega^{\omega}$-accelerated squares?

let $\{s_n\}_{n=0,1,2...}$ be a strictly increasing sequence of positive integers, indexed by the finite ordinals. if $S$ denotes the space of all such sequences, let the accelerator function, $\alpha ...
2
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0answers
131 views

Is there a nice characterization of posets induced by trees?

Define that a tree in $X$ is a set of ordinal-indexed sequences with codomain $X$ that is closed under the operations of restricting to an ordinal. (I do not know if this definition is standard.) ...
1
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40 views

How many equivalence classes in a set of well-orders of a set

The question: Let $S$ be a set and $\operatorname{wo}(S)=\{X: X\subseteq S \land (X,\le) \text{ is a well order}\}$. Furthermore, partition $\operatorname{wo}(S)$ into equvialence classes based on ...
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15 views

Ordinal-indexed exponentation

I have read from somewhere that $2^{\aleph_0}$ is uncountable. However, I have also read from somewhere else that $2^\omega=\omega$. Do the above statements contradict with each other? In ...
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60 views

$\epsilon_0$ is closed under addition, multiplication, exponentiation of ordinals

Question: Let $f: \Bbb{N}\rightarrow \text{Ord}$ (where "Ord" is the set of ordinals) be defined inductively by: $$f(0)=\omega\\ f(n^+)=\omega^{f(n)}$$ Let $\epsilon_0=\{\sup f(i) : i \in \Bbb{N}\}.$ ...
1
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0answers
62 views

Subset of rational numbers isomorphic to $\omega^{\omega+1}$

Does anyone know how to find/prove a rational subset isomorphic to $\omega^{\omega+1}$? I have couple of ideas but I find it hard to prove.
1
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29 views

Cantor-Bendixson rank of a first countable space

This question has been bothering me for quite a while, so let me ask it here. Is there a first-countable compact space $X$ with uncountable Cantor-Bendixson index? By a Cantor-Bendixson index I ...
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0answers
62 views

Ordinal Numbers within Simple Type Theory

A little preamble: I recently came across the system of higher-order logic known as simple type theory, and I was immediately intrigued by it, as it seemed to be exactly what I was looking for as an ...
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0answers
36 views

Derivatives of ordinal order

This question actually arises from this answer to another question, which contains the sentence A function is smooth is it has derivatives of infinite order. While the author surely didn't ...
1
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0answers
43 views

Existence of increasing pair of labeled trees in an infinite sequence

Assume labeled rooted trees with labels from a fixed set $\{1\ldots m\}$. For a tree $T$, we have: $V(T)$ the set of vertexes, $root(T)$ the root of the tree, $l_T: V(T)\rightarrow \{1\ldots m\}$ ...
1
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0answers
47 views

A question regarding Silver's Theorem

I am reading Jech's book Introduction to Set Theory. As an introduction to Silver's theorem, it is stated that: If $\aleph_\alpha$ is a strong limit cardinal and if $k < \aleph_\alpha$ and $\...
1
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0answers
25 views

Exponentiation on order types

How is exponentiation defined on order types? We know that $2^\omega=\omega$. What is $2^{\omega^*}$? Is it $\omega^*$? $\eta$? $\lambda$? I'm guessing $\eta$, but I'm not sure. $\omega$ is the ...
1
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0answers
63 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 $\...
1
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0answers
63 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
52 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 ...
1
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0answers
165 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 +1)=\...
1
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0answers
56 views

ind-completion and ordinals

Given a category $C$, we have its ind-completion $Ind(C)$ whose objects are filtered diagrams in $C$. Assuming the axiom of choice, is any object in $Ind(C)$ isomorphic, in $Ind(C)$, to an ordinal ...
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59 views

$\sigma$-field and Uncountable ordinal

I have been trying to get my head around this question. Any help greatly appreciated. Let $\mathscr{C}$ be any class of subsets $\Omega$ with $\emptyset,\Omega\in \mathscr{C}$. Define $\mathscr{C}_0=\...
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0answers
75 views

Generalization of simple and transfinite induction

Definition For a set of ordinals $\boldsymbol\alpha$ and ordinals $\gamma$, $\beta$, let $$\boldsymbol\alpha \xrightarrow[\gamma]{}\beta$$ symbolize the proposition that $(\boldsymbol\...
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37 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 $\...
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175 views

Relationship between ordinals and rank of well founded relations on $\mathbb N$

I want to understand the relation between ordinals and well founded relations on $\mathbb N$. I found a nice starting point here cut-the-knot/ordinals. Ordinals start like this 0={}, 1={0}, 2={0,1}, ...
0
votes
0answers
37 views

Any initial infinite ordinal is of the form $\omega_{\alpha}$

Let $\gamma$ be an initial infinite ordinal. Show that $\gamma=\omega_{\alpha}$ for some ordinal $\alpha$ The definition I use for $\omega_{\alpha}$ as follows: Suppose $\omega_{\beta},\aleph_{\beta}...
0
votes
0answers
17 views

Ordinals defined by successor and union

Let $M$ be the smallest set of ordinals satisfying $0\in M$ $x\in M\implies x+1\in M$ $S\subseteq M\implies \bigcup S\in M$ What does $M$ look like? Is it all ordinals? Is it all countable ...
0
votes
0answers
14 views

Order type of a sum ($\bigcup$) of sets

A quick question. Is $$\textrm{ot}(\bigcup\limits_{\gamma <\lambda}\alpha_{\gamma})=\bigcup\limits_{\gamma <\lambda}\textrm{ot}(\alpha_{\gamma})?$$ where $\textrm{ot}$ stands for the order type (...