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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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 ...
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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 ...
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39 views

Well ordering of type epsilon one

I have been very interested in the countable ordinals for awhile now, but one thing has eluded me despite my research into the subject. What is a well-ordering of the natural numbers corresponding to ...
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113 views

Bijection between countable ordinals and reals

The set of all countable ordinals is $\omega_1$, which has a cardinality of $\aleph_1$. When accepting the continuum hypothesis, $2^{\aleph_0} = \aleph_1$, so a bijection between countable ordinals ...
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Is the ordinal $\omega \uparrow^\omega \omega$ still recursive?

In this question, a very large countable ordinal $\omega \uparrow^\omega \omega$ is defined. Is this ordinal still recursive?
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79 views

Order type of standard models of arithmetic

The standard model of PA has order type $\omega$. By compactness PA has a model of order type $\omega+n$ for any $n$, since every finite subset of the following set of statements is provable: ...
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38 views

Every woset is an ordinal.

I use this notation: a well ordered set $Y$ is an ordinal if for every $a\in Y$, $Y_a=a$, where $Y_a=\{y\in Y|y< a\}$. Now, I know that for every woset there is an isomorphism from that woset to a ...
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2answers
50 views

Noninital ordinal of a set

If $\mathbb R$ is well ordered by an order $\ll$ such that the ordinal of R and that order $\alpha$ is not initial. How can I show that there is a closed interval $[a,b]$ ($a<b$) such that the ...
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109 views

Godel's pairing function and proving c = c*c for aleph cardinals

I have a few questions about Godel's pairing function and proving that c = c * c for aleph cardinals. Mostly, though, I'm concerned that most of the proofs I've seen are erroneous, and this concerns ...
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134 views

Understanding countable ordinals (as trees, step by step)

Even though ordinal numbers – considered as transitive sets – are perfect non-trees, it is worth (and natural) to visualize them as trees, starting from the finite ones which are given as ...
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136 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 $\omega^\omega$: ...
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Why $\omega+1$ and $\omega^2$ are not cardinal numbers?

I see the following definition of cardinal number in notes: An ordinal $\alpha$ is a cardinal number if $|\beta|<|\alpha|$ for all $\beta\in\alpha$. Why $\omega+1$ and $\omega^2$ are not cardinal ...
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For all infinite cardinals $\kappa, \ (\kappa \times \kappa, <_{cw}) \cong (\kappa, \in).$

I don't understand the proof to the a/m claim. How we know that $\eta < \kappa$ and $(\alpha,\beta)<_{cw} (0, \eta) $ and hence $h: \mu \to \eta \times \eta$ is injective? Appreciate if anyone ...
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50 views

Naturally definable order relations

Each and every function $f:X\rightarrow Y$ between two totally ordered sets $(X,\leq_X), (Y,\leq_Y)$ induces a relation $\preceq$ on $X$ by $$x \preceq x' :\equiv \begin{cases} x &\leq_X x' ...
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181 views

How uncountable is the set of countable ordinals?

From the answers (and comments) to this question on the uncountability of countable ordinals I don't get a lucid picture: How can I see that the uncountability of the set of countable ordinals ...
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43 views

If $\alpha,\beta$ are at most countable, so are $\alpha+\beta,\alpha\cdot\beta,\alpha^\beta$

If $\alpha$ and $\beta$ are at most countable ordinals, prove the following are at most countable: $\alpha + \beta$ $\alpha \cdot \beta$ $\alpha^{\beta}$
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34 views

A problem on ordinal arithmetic.

Let $0 < \alpha \leq \beta$ be ordinals. Could anyone advise me on how to prove there exists $\delta $ and $\xi$ such that $\xi < \alpha$ and $\alpha \ . \delta + \xi = \beta?$(ordinal ...
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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& ...
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55 views

Reducing ordinals in the representation of a limit ordinal as $\bigcup_{\beta < \alpha} \beta$

Let $\alpha$ be a limit ordinal such that $\kappa$ < $\alpha$ < $\kappa^{+}$ where $\kappa$ is initial ordinal (so $|\alpha| = \kappa$). I want to know whether or not I can find a sequence ...
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47 views

Prove that for any $m, n \in\omega$ that (i) $m + 0 = m$, (ii) $m + n^+ = (m + n)^+$

I am getting repeatedly lost trying to approach this question: Prove that for any $m, n \in\omega$ that (i) $m + 0 = m$, (ii) $m + n^+ = (m + n)^+$ I can fairly well grasp the idea that the ...
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157 views

What are ordinal numbers? [closed]

I know about the cardinal numbers and infinite cardinals and I know that Cardinals and ordinals are somehow related, but I just don't understand how ordinals work What exactly are ordinals and how ...
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73 views

Uncountable compact sets in ordinal topology

The question I am interested in is the following: Let $\omega_1$ be the first ordinal with an uncountable number of predecessors. We consider $X=[0,\omega_1]$ supplied with order topology. Prove ...
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1answer
64 views

A representation of well-orderings?

Is there a well-ordering $P$ of the set of real numbers $\mathbb{R}$ such that there is NO function $f: \mathbb{R}->\mathbb{R}$ satisfying the property: for all $x,y \in \mathbb{R}$, $xPy$ iff ...
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143 views

Let $\gamma $ be an ordinal. Prove there is an ordering preserving $f:\gamma \to \mathbb{R}$ iff $\gamma < \omega_1.$

Let $<$ be the usual ordering on $\mathbb{R}.$ If $\gamma$ is an ordinal, then $f:\gamma \to \mathbb{R}$ is ordering preserving if $\forall \alpha \in\gamma \forall \beta \in \gamma [\alpha \in ...
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38 views

Existence of certain $\left\langle{\alpha_n | n \in \omega}\right\rangle$

Let $\beta$ be a countable limit ordinal. Prove $\exists$ sequence $\left\langle{\alpha_n | n \in \omega}\right\rangle$ with the following properties: $(1): \alpha_0 = 0\;;$ $ (2): \forall n \in ...
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56 views

Functions from ordinals to ordinals

I'm trying to solve this problem, which appears in Schimmerling's "A Course in Set Theory." Problem. Find two functions $$f:\omega\rightarrow\omega\cdot2$$ and ...
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101 views

Proving, for every three ordinals $\alpha,\beta,$ and $\gamma$, that $\alpha^{\beta+\gamma}=\alpha^{\beta}\alpha^{\gamma}$

I am trying to prove, for every three ordinals $\alpha,\beta,$ and $\gamma$, that $\alpha^{\beta+\gamma}=\alpha^{\beta}\alpha^{\gamma}$. Proof is by induction on $\gamma$. Basis: If $\gamma = 0$, ...
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87 views

$\alpha > \omega$ is an $\epsilon$-number iff $\beta^\gamma < \alpha, \forall \beta, \gamma < \alpha$

Prove: $\alpha > \omega$ is an $\epsilon$-number iff $\beta^\gamma < \alpha, \forall \beta, \gamma < \alpha$. From left to right: if $\omega >\beta: \alpha=\omega^\alpha> ...
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1answer
105 views

Ordinals: if $\alpha < \omega^{\beta}$ then $\alpha + \omega^{\beta} = \omega^{\beta}$

I'm trying to proof that if $\alpha < \omega^{\beta}$ then $\alpha + \omega^{\beta} = \omega^{\beta}$, where $\omega$ is the least infinite ordinal. I started with transfinite induction on ...
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1answer
57 views

How to compute a Von Neumann Successor - ∪∪7?

I wonder if I have taken the correct approach to demonstrating what ∪∪7 is, using Von Neumann's proposal? And if I am indeed correct? My answer appears a little odd to me! Question: What is ∪∪7? By ...
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Additively indecomposable ordinals

As a continuation to my question here: Ordinal inequalities. I have tried to write down a proof for this claim: If $\alpha$ is of the form $\omega^\delta$ for some ordinal $\delta$, then, For every ...
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75 views

Ordinal inequalities

I have a problem with the following exercise: First a definition: $ \mathord{\in} = \{\langle x,y \rangle; x \in y \}$ Exercise: Show that the following conditions are equivalent: a. For every ...
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195 views

Showing ordinal addition and ordinal multiplication are defined

I've been reading absoluteless results in Kunens't latest Set Theory text. After talking about $\Delta_0$ formulas and absoluteness, he mentions that certain concepts are absolute, but not $\Delta_0$. ...
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34 views

Can ordinals multiplication be defined by recursion?

I am trying to figure out weather ordinals multiplication be defined by transfinite recursion. It seems a bit problematic to me since multiplication is a function $f: \alpha \times \beta \rightarrow ...
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85 views

the Class of Ordinals is a set?

Why the class of ordinals On is not a set? Please dont tell me it's well-ordered till you tell me what you mean by an ordering on something that you dont know in first place if it is a set or not.
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$f: \omega_1 \to \omega_1$. $\forall$ $\alpha$ $\exists$ $\beta > \alpha $ with $f(\beta)=\beta $.

Prove: if $f: \omega_1 \to \omega_1$ is an increasing, continuous, unbounded, function, then $\forall$ $\alpha$ $\exists$ $\beta > \alpha $ with $f(\beta)=\beta $. Can anyone give me a tip?
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If $A$ is a countable subset of $\omega_1$, then there is an $ \alpha < \omega_1$ such that $A \subseteq \alpha$

Prove : if $A$ is a countable subset of $\omega_1$. Then there exists $ \alpha < \omega_1$ with $ A \subseteq \alpha$. I don't really know where to start, can anyone give a tip first?
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62 views

An ordinal between an ordinal and its cardinality

Suppose $\alpha, \beta$ are ordinals (the first is not a cardinal) and $|\alpha| \subset \beta \subseteq \alpha$. Then why $|\alpha| \sim \beta?$ I know Cantor–Bernstein theorem does the job ...
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about definition of a cardinal

Definition: the cardinality of a set $A, |A|$ is the least ordinal s.t. $A \sim \alpha$ Definition: We define a cardinal to be an ordinal $\alpha$ s.t. $\alpha = |\alpha|.$ i.e, an ordinal s.th. ...
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52 views

Distinguishing characteristic and spurious properties

When working in set theory with definitions of somehow primitive notions like ordered pairs (Kuratowski) or ordinal numbers (von Neumann) one is supposed to use only the characteristic properties of ...
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41 views

$\bigcup X$ = $\sup(X)$ for a set $X$ of ordinals

Let $X$ be a set of ordinals. Prove that $\bigcup X$ = $\sup(X)$(which is the least ordinal $\ge$ every ordinal in $X$).
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56 views

An ordinal is well-ordered by inclusion

Recall that we define $\alpha < \beta$ if $\alpha \in \beta$ and $\alpha \leq \beta$ if $\alpha \in \beta$ or $\alpha = \beta$. Prove that for ordinals $\alpha, \beta$, $\alpha \leq \beta$ iff ...
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40 views

Showing that intersections are not defined

I'm checking to see why intersections are not defined when looking at the class $A$ defined by: $$ A = ON \cup [ON]^2\;,$$ where $ON$ is the class of ordinals and $[ON]^2$ the class of unordered ...
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43 views

Set of ordinals less than a given ordinal

Can anyone enlighten me about how this set can be well-ordered? That is, given any ordinal $\alpha$ the set of all ordinals less than $\alpha$, $W(\alpha)$, is well-ordered. I can show that this set ...
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Under what circumstances does the least $x \in X$ such that [whatever] still have this property when mapped into $Y$?

Let $X \lesssim Y$ denote well-ordered sets and $f : X \rightarrow Y$ denote the unique such order embedding whose image is downward-closed. Then its not necessarily the case that for all ...
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2answers
56 views

Euclidean algorithm for ordinals

I am trying to prove the Euclidean algorithm for ordinals. More specifically: For any ordinals $\alpha, \beta$ where $\beta >0$, there are unique $\gamma, \delta$ such that $\alpha = ...
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1answer
61 views

How would you “count” $\omega^\omega$

$\omega^\omega$ can be seen as the limit of $\omega^n$ which are all countable sets, and is thus countable. For the latter sets, there is an "easy" way list the elements out, but how would you do it ...
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1answer
94 views

Is the Church-Kleene Ordinal describable with Kleene's $O$?

Kleene's $O$ is an ordinal notation system that uses certain natural numbers to represent transfinite ordinals. It is a recursive notation system (although it's not decidable whether a number ...
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1answer
45 views

2-well-orderings on the same set

Can you find a set $X$, and two well-orderings $<_{1}, <_{2}$ of $X$ such that $(X, <_{1})$ is not isomorphic to $(X,<_{2})$. I know $X$ cannot be finite because if it were finite then ...
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74 views

Is ordinal analysis a non-recursive project?

A recursive ordinal is an ordinal that is the order-type for some recursive relation (i.e. a recursive well-ordering). We can represent recursive ordinals as natural numbers using Kleene's $O$, an ...