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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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
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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 ...
2
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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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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 ...
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67 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
141 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
148 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
152 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
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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
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180 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 ...
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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) ...
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1answer
108 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, ...
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1answer
235 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?
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2answers
117 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!
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1answer
60 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
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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 ...
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1answer
93 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
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1answer
80 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
102 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 ...
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2answers
264 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 ...
4
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1answer
299 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 ...
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226 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
80 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?
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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
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1answer
205 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
126 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
votes
2answers
81 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
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1answer
86 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
104 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
178 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. ...
4
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2answers
122 views

If $\alpha$ is an ordinal, proving that $\alpha\cup\{\alpha\}$ is an ordinal.

I refer to pg.4 of this article. Assuming $\alpha$ is an ordinal, we have to prove $\alpha\cup \{\alpha\}$ or $\alpha +1$ is an ordinal. Isn't this obvious from the construction of ordinals? As ...
2
votes
1answer
139 views

Why $\omega$ can't be bijectively mapped to $\omega +1$

Let $\omega$ be the order type of the totally ordered set $\mathbb{N}$, and $\omega +1$ the set $\Bbb{N}\cup \{0\}$. $0$ is greater than all the natural numbers as per this ordering. My question ...
7
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3answers
238 views

How to prove the Milner-Rado Paradox?

For every ordinal $\alpha<\kappa^+$ there are sets $X_n\subset\alpha$ $(n\in\Bbb{N})$ such that $\alpha=\bigcup_n X_n$ and for each $n$ the order-type of $X_n$ is $\le\kappa^n$. [By ...
2
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0answers
68 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.) ...
3
votes
1answer
89 views

Does every countable subset of the set of all countable limit ordinals have the least upper bound in it?

I'm sorry if the question is that kind of trivial, I just feel uncertain about these ordinals all the time. Is the answer to the following question "yes": Denote by A the set of all countable limit ...
0
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1answer
118 views

The limit elements of well-ordered proper classes

For any well-ordered class $W$ with no maximum element, we can define a successor function $f_W : W \rightarrow W$ by asserting that $f_W(w) = \mathrm{min}\{x > w \mid x \in W\}.$ This allows us to ...
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1answer
77 views

Statements regarding ordinal numbers [duplicate]

Let $m$ and $n$ be infinite ordinal numbers Which of the following is true a) $m<n \Rightarrow |m|^{|m|}<n^{|n|}$ b) $m+n$= Max{$m,n$} c) $m=n \Rightarrow |m|=|n|$ d)$|m|=|n| \Rightarrow ...
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votes
2answers
79 views

Why is $n+\omega=\omega$ for finite $n$?

I am trying to understand the following claim; $n+\omega=\omega$ where $n$ is order type of a finite set and $\omega$ is the order type of $\left\{ 1,2,\dots, \right\}$ with the usual meaning of ...
2
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1answer
82 views

Are the closed ordinals (apart from $0$ and $1$) precisely the regular cardinals?

Given a partially ordered set $P$, a collection of partially ordered sets, call it $\mathcal{Q}$, and an arbitrary function $f : P \rightarrow \mathcal{Q},$ we can form a new partially ordered set ...
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1answer
120 views

Cardinal numbers

Suppose $m, n$ are infinite ordinal numbers. $$a) m=n → |m|=|n|$$ $$b)|m|=|n| →m=n$$ $$c)m<n→ |m|<|n|$$ $$d)|\max{(m,n)}|< |m|+|n|$$ $$e)|m|<|n| →|m|^{|n|}<|n|^m$$ Which of the above ...
3
votes
3answers
118 views

Set $A$ has the cardinality of $\aleph_0$. Prove some properties of partial order $ \langle \mathcal P (A), \subseteq \rangle$.

I try to solve the following task: Set $A$ has the cardinality of $\aleph_0$. The truth is that in the partial order $ \langle \mathcal P (A), \subseteq \rangle$: (answer true or false) a) ...
2
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2answers
67 views

Why are the ordinal operations defined that way?

If $\mu$ is a limit ordinal, then we define that $$\alpha + \mu = \sup_{\beta < \mu} (\alpha + \beta).$$ Why don't we similarly define that if $\lambda$ is a limit ordinal, then $$\lambda + \beta ...
2
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3answers
323 views

show that every continuous real-valued function defined on $S_{\mathbb{\Omega}}$ is eventually constant

show that every continuous real-valued function defined on $S_{\mathbb{\Omega}}$ is eventually constant.Where $S_{\mathbb{\Omega}}$ denote the first uncountable ordinal. There is a hint that for each ...
3
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2answers
134 views

The concept of ordinals

I am trying to understand this concept and have some difficulties. For example, can I say that $\alpha$ is the cardinality of $\{1,2,3,...\}=\Bbb N$? And if so, what is $\alpha +1$? I guess it is ...
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1answer
76 views

This set of ordinals is $\in$-transitive?

I was reading this problema about ordinals. I'm following the Z.F theory. Let $A$ be a nonempty set of ordinals. Prove that if $\cup A \notin A$ then $\cup A$ it's a limit ordinal, i.e $\cup A $ is ...
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3answers
56 views

Let $\alpha$ be an ordinal then $\alpha=\cup\alpha$ or $\alpha=s(\cup\alpha)$.

Let $\alpha$ be an ordinal then $\alpha=\cup\alpha$ or $\alpha=s(\cup\alpha)$. Attempt: Since $\alpha$ is transitive $\alpha=\cup\alpha$ (here we are done) or $\alpha\supsetneq\cup\alpha$. If ...