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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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
58 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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45 views

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 ...
0
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1answer
78 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 ...
3
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1answer
199 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 $\Delta_0$. ...
2
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1answer
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 ...
0
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2answers
91 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.
3
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70 views

$f: \omega_1 \to \omega_1$. $\forall$ $\alpha$ $\exists$ $\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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42 views

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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2answers
63 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 ...
2
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3answers
46 views

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. ...
2
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1answer
54 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 ...
2
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2answers
44 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 $X$).
1
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1answer
58 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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1answer
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 ...
0
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0answers
45 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 ...
2
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2answers
58 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 = ...
2
votes
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 ...
2
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1answer
98 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 ...
0
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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 ...
4
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1answer
75 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 ...
2
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2answers
68 views

Is the following claim true: “every ordinal has the empty set as one of its elements”

Since we can encode every element as a set, suppose that we consider only sets in which every element is also a set. My question is, is it true that every ordinal has the empty set as one of its ...
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1answer
159 views

Long line is connected and compact

How to prove that the long line is connected and compact. I was trying to prove connectedness using contradiction but couldn't.
5
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1answer
152 views

What is the proof-theoretic ordinal of the first-order theory of real closed fields?

I recently asked a question on MathOverflow, concerning a predicative second-order theory of real numbers. Now the standard way of developing predicativity in the case of second-order arithmetic is ...
3
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2answers
154 views

Two exercises on set theory about Cantor's equation and the von Neumann hierarchy

Good evening to all. I have two exercises I tried to resolve without a rigorous success: Is it true or false that if $\kappa$ is a non-numerable cardinal number then $\omega^\kappa = \kappa$, where ...
2
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1answer
57 views

A function on ordinals.

Denote by $L$ the set of all non-zero countable limit ordinals. Is there an injection $f\colon \omega_1\times L\to L$ such that $f(\alpha,\beta)>\beta$ for all $(\alpha,\beta)$ in the domain of ...
4
votes
1answer
78 views

Other criteria for a set to be an ordinal.

I'm reading up on ordinals, and I've come across various definitions for what it means to be an ordinal. The main book that I'm following is Kunen's Set Theory book. He defines an ordinal as a set ...
5
votes
3answers
188 views

A few questions about “small” transitive sets.

I'm currently reading about ordinals, working on various exercises since I haven't formally studied the material before. I'm looking at transitive sets. A set $z$ is said to be transitive if and only ...
6
votes
1answer
79 views

Are there any nontrivial doubly-well-ordered sets?

A set $A$ is well-ordered (by an ordering $R$) if there is an $R$-minimal element in every nonempty subset of $A$. Call $A$ doubly well-ordered (by $R$) if $R$ well-orders $A$ and $R^c$ (converse) ...
1
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1answer
110 views

Construction of an infinite number type and other ideas

Construct an infinite number(?) that has a beginning, an infinite middle, and a end; such as 1000...0001, or 98111...1114 etc. Has this type of number been explored? Under some simple multiplications, ...
0
votes
1answer
255 views

First uncountable ordinal number

Why is $\Omega$ the first uncountable ordinal number and not $\omega^\omega$? Isnt the latter a countably infinite product of countably infinite sets and hence uncountable?
2
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2answers
123 views

Who is Epsilon ($\epsilon$)? a binary relation?

I was reading: Transitive set ordered by epsilon and http://www.princeton.edu/~jburgess/PHI323S13Problems.pdf (ex. 4) so, who is epsilon? Thanks in advance!
0
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1answer
63 views

Second element of fundamental sequence of Zeta-nought

I'm attempting to evaluate the second element of the fundamental sequence for the infinite ordinal known as zeta_0 zera_0, is defined as the supremum of: {epsilon_0, epsilon_(epsilon_0), ...
2
votes
2answers
119 views

detecting cofinality in ordinals

Suppose K is an unbounded well ordered set, and suppose K is minimal in the sense that each proper initial segment of K has strictly smaller cardinality. Suppose J is an unbounded well ordered set of ...
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0answers
15 views

$\Sigma$-product and Tychonoff product [duplicate]

I want to say that the below example is countable compact but not closed, $X$ a Tychonoff product $[0,1]^{ω_1}$ and as $Y$ the $\Sigma$-product $$ \{ x ∈[0,1]^{ω_1}:|\{α<ω_{1} :x(α)≠0 \}|≤ω ...
7
votes
1answer
94 views

What classification of countable ordinals above $\omega_1^{CK}$ exists?

What classification of countable ordinals above the Church–Kleene ordinal $\omega_1^{CK}$ exists? Are there such things as $\omega_2^{CK}$, $\omega_{\omega_\omega^{CK}}^{CK}$ or ...
1
vote
1answer
82 views

How to prove that $\text{pred}_{(A,\prec_A)}(y)=\{x\in A \,|\, x\prec_A y\}$

In Schimmerling's book on set theory, question 3.1 reads: "Let $(A,\prec_A)$ be a wellordering such that $A\neq\emptyset$. for each $y$ define $\text{pred}_{(A,\prec_A)}(y)=\{x\in A \,|\, x\prec_A ...
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1answer
104 views

The classification of order types

In response to a previous question which I asked concerning the order type of the Rationals, a reference was made (by MDJ) to a theorem of Cantor stating that any two countable, dense, linearly ...
5
votes
2answers
278 views

The order type of the rationals.

Herewith another mind-numbingly naive question from a reader of philosophy. My question concerns the order type of the rational numbers. Omega squared seems a natural first choice, but obviously ...
5
votes
1answer
341 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 ...
14
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2answers
229 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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1answer
82 views

$X = [ 0,\omega_1 )$ is maximal countably compact

(a) : why $X = [ 0,\omega_1 )$ is a first countable space? ($\omega_1$ is the first uncountable ordinal number.) (b) : is $X = [ 0,\omega_1 )$ maximal countably compact?
6
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0answers
111 views

AC iff $P(\delta)$ can be well-ordered for all $\delta\in{\bf On}$ [duplicate]

I'm trying to do the following exercise: Exercise. Assume ZF. Then AC is equivalent to the statement that $P(\delta)$ can be well-ordered for all $\delta\in{\bf On}$. I'm struggling with the ...
6
votes
1answer
207 views

Ordinal arithmetic exercise in Kunen

I'm self studying Kunen (2013 edition) right now, and got stuck on this exercise - I generally lack the tools and/or intuition for these kind of ordinal arithmetic exercises. Exercise. Let ...
2
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2answers
130 views

I read that ordinal numbers relate to length, while cardinal numbers relate to size. How do 'length' and 'size' differ?

I read that ordinal numbers relate to length, while cardinal numbers relate to size. How do 'length' and 'size' differ? Note : I am an absolute novice, and I'm having a little trouble visualizing ...
0
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2answers
67 views

How can we describe different sorts of limits using ordinals, and what kind of “work” can they do in transfinite induction?

I recently encountered transfinite induction and ordinal numbers for the first time in a real analysis text (Bruckner, Bruckner, and Thomson) and I am still coming to grips with these tools and their ...
2
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2answers
82 views

Definition of ordinals in Grothendieck toposes

What is a useful definition of an ordinal in Grothendieck toposes. Useful in this context means that I would like to construct fixed points of monotone maps on complete lattices using iteration from ...
4
votes
1answer
89 views

If $(W,<)$ is a well-ordered set and $f : W \rightarrow W$ is an increasing function, then $f(x) ≥ x$ for each $x \in W$

I could use a hand understanding a proof from Jech's Set Theory. Firstly, note that Jech defines that an increasing function $f : P \rightarrow Q$ is a function that preserves strict inequalities ...
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2answers
105 views

Prove that if $x,y\in\alpha$, then $x<y,x>y$ or $x=y$.

I refer to pg.4, Lemma 12 of this article on ordinal numbers. It says "If $x,y\in \alpha$, then $x<y, x>y, \text { or }x=y$". My question: As $\alpha$ is an ordinal, we know $x<\alpha$ ...
7
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3answers
180 views

Is there any countable ordinal number which has a member undefinable?

Let $Q\colon=\{\alpha \in Ord \mid \exists \beta \in \alpha(\beta \text{ is undefinable in } \langle\alpha,<\rangle )\}$ denote the set of all ordinal numbers which has an undefinable member. ...