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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Cantor-Bendixson rank of a first countable space

This question is bother me for quite a while, so let me ask it here. Is there a first-countable compact space $X$ with uncountable Cantor-Bendixson index? By a Cantor-Bendixson index I mean the ...
3
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2answers
49 views

Countable ordinal

I haven't done ordinal algebra for a long time and I can't remember how to prove that $\omega + \omega$ is a countable ordinal. Precisely what is the bijection between $\omega$ and $\omega \cdot 2$ ? ...
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1answer
20 views

Union of a chain of cardinalities?

I was trying to understand the union of a chain of cardinalities and I found this equation $$\kappa=\bigcup_{\alpha<\kappa} \alpha$$ for any cardinal $\kappa$ in the answers to this question. Can ...
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1answer
62 views

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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2answers
92 views

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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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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65 views

For which countable successor ordinals $\alpha$ is the reverse order isomorphic to the ideals of a PID ordered by inclusion?

Let $\alpha$ be a countable successor ordinal and $\alpha^{\mathrm{op}}$ the reverse order. For which $\alpha$ is there a commutative principal ideal ring $R$ such that the ideals of $R$ form a chain ...
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178 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
20 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 ...
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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 ...
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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 ...
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, ...
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2answers
137 views

What is the cardinality of the open ordinal space $[0,\Omega)$ if we remove neighborhoods of each limit ordinal?

Let $\Omega$ be the first uncountable ordinal with the order topology. For each limit ordinal $\lambda < \Omega$ let $U_\lambda$ be an open neighborhood of $\lambda$. What is the cardinality of ...
3
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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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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 ...
3
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2answers
269 views

Is there a categorification of (infinite) ordinal arithmetic?

Background for the curious reader: An ordinal $\beta$ is a transitive set in the sense that $\alpha\in\beta$ implies $\alpha\subset\beta$. Any ordinal is naturally well-ordered under $\in$ (so any ...
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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? ...
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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
229 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 ...
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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, ...
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1answer
92 views

How many omegas are there in $\large f_{\varepsilon_0}$?

For a description look at fast growing hierarchy at wikipedia. $\large f_{\varepsilon_0}$ is not defined any more, it is a power tower of omegas, but how many omegas ? I found a defition $$\large ...
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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
77 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 $R\subseteq A\times A$. ...
7
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1answer
272 views

what simple extensions can be naturally embedded into Conway's algebraic closure of $\mathbb{F}_2$?

John Conway proved in his book, On Numbers and Games (ch6, theorem 49) that the set of all ordinals smaller than $\omega^{\omega^\omega}$ form a field of characteristic 2 that is isomorphic to ...
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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 ...
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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
104 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
83 views

A question on transfinite induction

The following were 2 problems given to us on transfinite induction, The transfinite induction I saw on books and books etc was using ordinals but the definition we were given is different and I still ...
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6answers
814 views

Principle of Transfinite Induction

I am well familiar with the principle of mathematical induction. But while reading a paper by Roggenkamp, I encountered the Principle of Transfinite Induction (PTI). I do not know the theory of ...
10
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4answers
3k views

Examples of transfinite induction

I know what transfinite induction is, but not sure how it is used to prove something. Can anyone show how transfinite induction is used to prove something? A simple case is OK.
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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
50 views
1
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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 ...
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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 ...
3
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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 ...
0
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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 ...
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2answers
56 views

Theorem $($Transfinite Induction and Construction$)$ (James Dugundji - 5.1, Page - 40)

Let $W$ be a well ordered set, and let $Q \subset W$. If $[ W(x) \subset Q] \Rightarrow [ x \in Q]$ for each $x \in W$, then $Q = W$. For each $a \in W$, the set $W(a) = \{x \in W; (x\prec 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 ...
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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
76 views

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 ...