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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Elementary Set Theory: Ordinals, Well Orderings and Isomorphisms

I need to show that for any countable ordinal $\alpha$ there is a set A $\subseteq \mathbb{Q}$ such that (A, <) is isomorphic to ($\alpha, \in$). To do it I am supposed to show the following ...
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Do fett ordinals exist?

Define a class $K$ of ordinals inductively as follows: $0=\emptyset\in K$. For all $\alpha\in K$, the succesor of $\alpha$ is also an element of $K$. For every function $f\colon \mathbb N\to K$, the ...
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0answers
17 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 ...
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1answer
25 views

Can't the first (/second) transinfinite ordinal replace the first (/second) uncountable ordinal in several counterexamples?

An example of a sequentially compact but not compact space is $\omega_1$. Indeed, any sequence of countable ordinals either has infinite elements below some countable ordinal, or has an ascending ...
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2answers
170 views

Can every set be expressed as the union of a chain of sets of lesser cardinality?

If a set $S$ has countably many elements $\{x_n\}$, it can be expressed as a union of a chain of finite sets $$ \{x_1\} \subset \{x_1,x_2\} \subset $$ But what about a set of arbitrary cardinality ...
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1answer
47 views

Order type of final segment of countable limit ordinal

Suppose $\gamma$ is a countable limit ordinal, with only finitely many limit ordinals less than it. If $\zeta$ is the greatest limit ordinal less than $\gamma$, does this necessarily imply that the ...
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0answers
18 views

Commutativity of operations on ordinal numbers? [duplicate]

I read on the Wikipedia for ordinal numbers that the operations of addition and multiplication aren't commutative, more specifically: "addition and multiplication are not commutative: in particular $1 ...
3
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1answer
60 views

Isomorphic ordinals are equal

Let $\alpha$ and $\beta$ be two isomorphic ordinals. Then $\alpha = \beta$. I want to whether the following proof is correct. I already know that there are three prossible cases: $\alpha \in ...
2
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1answer
22 views

Equal cardinality implies isomorphism of ordering

I think I have proved this (should-be false) lemma. For any set $X$, if $X$ is equipotent to an ordinal $\alpha$, then $X$ can be ordered so that it is isomorphic to $\alpha$. This is the proof ...
4
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1answer
69 views

Is it true that $\lvert\alpha+\beta\rvert=\lvert\alpha\rvert+\lvert\beta\rvert$ for ordinals $\alpha$ and $\beta$?

Suppose $\alpha$ and $\beta$ are ordinals. I was asked to prove that $\lvert\alpha+\beta\rvert=\lvert\alpha\rvert+\lvert\beta\rvert$. However, I think to have found a counterexample for this equality, ...
0
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1answer
34 views

Ffind an ordinal type of the well-ordering

Let $x_n$ for $n \in N$ are variables of predicate logic. Let $F(L)$ defines set of all formulas of language $L$ and $Σ(L) = L ∪ \{(,), ¬, →, ∀\} ∪ \{x_n | n ∈ N\}$ defines corresponding alphabet. We ...
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1answer
37 views

What is $1^\omega$?

In Wolfram Mathworld, Ordinal exponentiation $\alpha^\beta$ is defined for limit ordinal $\beta$ as: If $\beta$ is a limit ordinal, then if $\alpha=0$, $\alpha^\beta=0$. If $\alpha\neq 0$ then, ...
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0answers
29 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}| = ...
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1answer
43 views

A question on proof of the Zermelo's theorem “Every set is well-orderable.”

There exists some proof for the theorem. Some of them use Transfinite Recursion. Some of them use same argument with the following. Proof:(copied from proofwiki) "Let $S$ be a set. Let ...
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2answers
66 views

List of limit ordinals [closed]

I am trying to understand ordinals and cardinals. Seeing a list of limit ordinals would be helpful. A small question outside of this, are the limit ordinals related to cardinals? In what way? ...
0
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1answer
46 views

Where is the mistake in this set-theoretic argument?

Let $\omega$ be the first infinite ordinal and for all $n\in \omega $ define $n=\{0,1,2,...,n-1\}$. In particular, $2=\{0,1\}$. Let $f:\omega\rightarrow \omega$, $f(x)=x^2 $. Now ...
3
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2answers
227 views

Any Set of Ordinals is Well-Ordered

Is this a theorem of ZF or a theorem of ZFC? My suspicion is that it is a theorem of ZFC (the proof I've seen in P. L. Clark's online notes on countable ordinals requires selecting an element from ...
3
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1answer
55 views

Is this a way to define an ordinal $\lambda$?

A set $\lambda$ is by definition an ordinal if $\forall x\in\lambda : x\subseteq \lambda$ and $\lambda$ is well-ordered by $\in$ . May I equally define an ordinal as a set $\lambda$ having ...
0
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1answer
40 views

Is model theory needed to understand ordinal logic?

By ordinal logic, I mean turing's ordinal logic. I'm going to learn first order logic, elementary set theory, basic computability, and godelian incompleteness as prerequisites for ordinal logic. But, ...
1
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1answer
35 views

Comparing Hartogs number to set of all sets of relations on a set

Let $A$ be a set and let $\mathcal{H}(A)$ be the Hartogs number of $A$. Show that $|\mathcal{H}(A)|\neq|\mathcal{P}(\mathcal{P}(A\times A))$. Proof attempt: By contradiction. Suppose that ...
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1answer
76 views

Surjective function into Hartogs number of a set

For any set $A$, there is a surjective function $f:\mathcal{P}(A\times A)\longrightarrow\mathcal{H}(A)$. $\mathcal{H}(A)$ is the Hartogs number of $A$. Proof attempt: Suppose $R\subseteq A\times A$. ...
0
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3answers
140 views

Best known upper and lower bounds for $\omega_1$

What are the best known lower and upper bounds for $\omega_1$? Are there sharper lower bounds than $\varepsilon_0$? And are there any known upper bounds which can be explicitly constructed like ...
2
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3answers
88 views

Is $\omega_0-1$ infinite?

I have read in another answer Is infinity an odd or even number? that the $\omega_0$ is the "smallest infinity", but is $\omega_0-1$ not also infinite?
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1answer
35 views

Vanishing Cantor-Bendixson derivative if and only if scattered

In the question asked here, it is claimed in a comment and answer of Brian M. Scott that a subset of $\mathbb{R}$ is scattered if and only if it has vanishing Cantor-Bendixson derivative. I thought ...
3
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1answer
101 views

What are all the finite transitive sets?

In set theory, we come across the definition of an ordinal as a well-ordered transitive set. And then we show that $\omega$ and all its elements are ordinals. My question whether there are finite ...
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1answer
75 views

Countable Ordinals

In an intro topology class we briefly brought up ordinal numbers during a conversation of transfinite induction. I believe I understand how the ordinal numbers work, at least up to $\omega^\omega$. ...
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0answers
50 views

(Set theory, Ordinal number) How to construct explicit isomorpism between epsilon(ordinal number) and subset of Q?

Now I know that there 'is' isomorpism between them, but the problem asks explicit isomopism. How can I do this? (This is problem from Introduction to Set Theory(Hrbacek,Jech) chapter6.)
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2answers
49 views

What would be an example of a well-ordering of the natural numbers of type $\omega^3$? [closed]

What would be an example of a well-ordering of the natural numbers of type $\omega^3$ ?
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1answer
43 views

Is it the case that for all ordinal numbers $a$ and $b$, $|a^b|=|a|^{|b|}$? [closed]

Is it the case that for all ordinal numbers $a$ and $b$, $|a^b|=|a|^{|b|}$?
1
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0answers
26 views

(Ordinal number)How to prove that epsilon-naught is countable without using Axiom of Choice? [duplicate]

How to prove that epsilon-naught is countable without using Axiom of Choice? or, Can we explicitly show that there is isomorpism between epsilon naught and subset of rational number?
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1answer
80 views

There is no infinite descending chain of ordinals

I'm currently reading a set theory book and it uses the fact that There is not infinite descending chain of ordinals. But I can't find the proof. (And also I remember Topology by Munkres uses ...
3
votes
1answer
30 views

Can ordinal numbers be generalized to arbitrary classes?

The following is an obvious potential, albeit incomplete, method of generalization. One class paired with an ordering $s$ would be of the same order type as another $t$ iff there exists an ...
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0answers
56 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
43 views

What axioms satisfy a limit ordinal?

Let us consider the pair $(\alpha,\in)$ where $\alpha$ is a limit ordinal. The question is what axioms of ZFC satisfies that pair. I think it satistifies all axioms but I'm not sure...
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1answer
51 views

Is there some result that says a theory cannot prove the consistency of any of its extensions?

Is there some result that says a (sufficiently strong) theory cannot prove the consistency of any of its extensions? Or something along these lines?? More generally, is there a result that says a ...
6
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1answer
68 views

Inconsistencies in ordinal arithmetic (normal form)

I'm trying to understand how to bring ordinals of the form $(\omega +n)\omega$ to normal form. Unfortunately, I am confused by what seem to be inconsistent answers to very similar questions: In a ...
1
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1answer
23 views

Proving basic fact about ordinals

In a set of online lecture notes, I saw the following proposition. Let $C$ be a set of ordinals. Then $\sup \left\{ \alpha +\beta:\beta\in C \right\} =\alpha +\sup C$. How can I prove this?
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3answers
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Let $\omega$ be the ordinal of the well ordered set $\mathbb{N}$, what does means $\omega^\omega$?

Let $\omega$ be the ordinal of the well ordered set $\mathbb{N}$, then we can define the ordered sum and product, and so the sum and product of ordinal. Is easy to see that ...
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0answers
31 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 ...
7
votes
0answers
107 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 ...
5
votes
2answers
50 views

How to prove the not-so-long rays are homeomorphic to the reals?

The long ray (half of the long line) is an interesting topological space. It is defined as the order topology on $\omega_1$$\times [0, 1)$ with lexicographic order. Basically, it is an uncountable ...
5
votes
2answers
37 views

How many limits of limits are there in $\omega_1$?

We know there are $\omega_1$-many limit ordinals less than $\omega_1$. What about ordinals that are limits of limit ordinals? Are there also $\omega_1$-many of these?
0
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1answer
69 views

How to view beth number as ordinal?

The following is the snippet of the paragraph given in my lecture note: (Note: $\beth$ is the beth-number.) (When I introduced $\beth_\alpha$ I told you that it was to be a particular size of ...
3
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0answers
84 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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2answers
77 views

Proving that $\langle \omega+1, \leq \rangle$ and $\langle \omega + \omega^*, \leq \rangle$ are not elementarily equivalent

This is an exercise from Kees Doet's Basic Model Theory. The basic idea is to show that an ordering of type $\omega + 1$ is not isomorphic to an ordering of type of $\omega + \omega^*$, where ...
2
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1answer
78 views

order of infinite countable ordinal numbers

I'm trying to understand ordinal arithmetic. If one had an ordered list of the some subset of countable ordinal numbers, what order would the following 6 countably infinite ordinals be in? If the ...
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0answers
136 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 < ...
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2answers
152 views

Cantor-Bendixson rank of a closed countable subset of the reals, and scattered sets

I am reading the notes in the following link, and I am a bit unclear about the connection between scattered sets, countable sets, and sets for which the Cantor-Bendixson derivative is eventually the ...
5
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1answer
111 views

The ordinals as a free monoid-like entity

Let $\kappa$ denote a fixed but arbitrary inaccessible cardinal. Call a set $\kappa$-small iff its cardinality is strictly less than $\kappa$. By a $\kappa$-suplattice, I mean a partially ordered set ...
2
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1answer
37 views

limit and infinite ordinals: same thing? [duplicate]

I'd like to clarify my understanding re: limit and infinite ordinals. This post says: An initial infinite ordinal is a limit ordinal. Is it true the other way around? That is if I have a limit ...