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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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
66 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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3answers
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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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1answer
55 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
50 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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1answer
63 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
42 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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0answers
48 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
69 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
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1answer
64 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
106 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
46 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
83 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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2answers
73 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
197 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
159 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
163 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
60 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
84 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 ...
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3answers
220 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
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1answer
82 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
113 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
283 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
127 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
72 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
129 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
100 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 ...
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1answer
88 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
119 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
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2answers
316 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
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2answers
478 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
242 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
87 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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0answers
114 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
221 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
142 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 ...
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2answers
69 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
89 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
100 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
106 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
194 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
144 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
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1answer
150 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
271 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
75 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
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1answer
99 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
136 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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votes
1answer
80 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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2answers
81 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 ...