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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630 views

Ordinal exponentiation - $2^{\omega}=\omega$

This is my understanding of ordinal arithmetic - two ordinals are the same as one another if there is an order-preserving bijection between them. So for instance $$1+\omega = \omega$$ because if $...
2
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2answers
112 views

$\omega_1^{CK} - \omega$ - infinite or finite set? And boundary

I am curious whether $\omega_1^{CK} - \omega$ would result in a finite set or infinite set. Does anyone know what happens? Edit: OK, let me add one more question: Suppose that we take $\omega \cdot \...
3
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1answer
101 views

What's the rank of this well founded relation?

Definition A tree is an ordered list of trees. (N.B these are finite objects and there is a very simple computable bijection of them with $\mathbb N$) Examples [] and [[],[],[]] and [[[]],[[],[],[]],[...
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0answers
175 views

Relationship between ordinals and rank of well founded relations on $\mathbb N$

I want to understand the relation between ordinals and well founded relations on $\mathbb N$. I found a nice starting point here cut-the-knot/ordinals. Ordinals start like this 0={}, 1={0}, 2={0,1}, ...
8
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1answer
192 views

what does it mean that constructible universe is definable from ordinals?

I know how constructible universe is created, but I also separatedly read that the universe is definable from ordinals - so I am wondering what it really means.
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1answer
34 views

Well-ordered family of sequences each decreasing faster than the earlier ones?

Let $\alpha$ be a countable ordinal. Is there necessarily a family $(\epsilon_{\beta,n})_{n\in\mathbb{N},\,\beta<\alpha}$ such that $\epsilon_{\beta,n}>0$, $\sum_n \epsilon_{\beta,n} = 1$, and $\...
1
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1answer
102 views

Disjoint unions of limited cardinality: is the following conjecture/proof correct?

Is the following conjecture about collections of sets correct? And if so, does the proof work? Conjecture. Let $\kappa$ denote an infinite cardinal number, and let $\mathcal{K}$ denote a collection ...
0
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1answer
159 views

Does there exist a set theory where the following non well-founded definition of the ordinals is possible?

I was wondering if there is a non well-founded set theory such that the following implementation of the ordinals is possible. $0 = \emptyset$ $1 = \{1\}$ $2 = \{1,2\}$ ... $\omega = \{1,2,3,\cdots\}$...
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1answer
67 views

Comparing how “high” the ordinals go in one model of ZFC versus another.

Given two transitive models of ZFC, is there a way of comparing how "high" the ordinals go in one model versus another?
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2answers
95 views

How to understand closed subsets of limit ordinal?

On page 20, Constructibility, K.J.Devlin, (Let $\alpha$ is a limit ordinal)A set $A \subseteq \alpha$ is closed, iff $\bigcup A \cap \gamma \in A$ for all $\gamma < \alpha$. Equivalently, if we ...
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2answers
849 views

Why are infinite cardinals limit ordinals?

My book states this as obvious, but it isn't so trivial to me. thanks
3
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0answers
310 views

How to derive Church-Kleene ordinal

Crossing-out: (How does one prove the existence of Church-Kleene ordinal? Also, why is it labeled as $\omega_1^{CK}$? And why is it first ordinal not hyperarithmetical, and is the first admissible ...
11
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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.
2
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1answer
108 views

open subsets of $\omega_1+1$

Let $ω_1$ is the first uncountable ordinal and $ω_1+1$ together with the order topology. Let $U$ and $V$ two disjoint uncountable set consisting of non-limit ordinals, then $U$ and $V$ are open and ...
4
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1answer
59 views

$\omega_1$ is $C^*$-embedded in $\omega_1+1$

consider the space $ω_1$ (the first uncountable ordinal) and $ω_1+1$ together with the order topology. is it true that $ω_1$ is $G_\delta$-dense subspace of $ω_1+1$? $\omega_1$ is $C^*$-embedded in $...
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1answer
42 views

How to compute immediate previous ordinal in von Neumann representation?

I don't know how to state the question more clear: why it is so hard to compute immediate previous ordinal in von Neumann representation? This is about Better representaion of natural numbers as sets?...
4
votes
2answers
394 views

Another basic Ordinal arithmetic question: $\omega \cdot (\omega+1)=?$

I am getting confused about ordinal arithmetic I am trying to figure out what $\omega \cdot (\omega+1)$ is. The fact that the distributive law applies from the left side makes me think that the ...
1
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1answer
255 views

Help with basic Ordinal arithmetic: what is $(\omega+2)\cdot \omega$

I am having trouble with some simple ordinal arithmetic. I am trying to figure out what $(\omega+2)\cdot \omega$ is. If you represent $\omega + 2$ as: $\omega+2 = 0_0 < 1_0 < 2_0 < ... < ...
2
votes
1answer
80 views

A set bijective with an ordinal

Lemma I.10.10 in Kunen's set thoery (2011) claims that a set $A$ is well-orderable iff $A$ is bijective with an ordinal. He does not prove this, and one direction is clear to me. However, I can't ...
6
votes
2answers
533 views

How to think about ordinal exponentiation?

I'm just trying to understand better how to see $\alpha^{\beta}$ for an arbitrary ordinal. I've already know that one can think about $\alpha . \beta$ as $\langle \alpha \times \beta, AntiLex\rangle$ ...
2
votes
2answers
221 views

Does the set of all ordinals strictly dominated by a given set exist in ZF?

How do I prove that $$ \{\alpha\in\mathsf{On}\,|\,\alpha\prec A\}\in V,$$ assuming $A\in V$? I know that if AC is assumed, this set is equal to $\mbox{card}(A):=\mu_\alpha(\alpha\approx A)$, and ...
5
votes
1answer
112 views

$\varepsilon$-numbers

An ordinal $\xi$ is an $\varepsilon$-number (where does this name come from?) if $\omega^\xi = \xi$. Is the set of all countable $\varepsilon$-numbers closed in $\omega_1$? In other words, does an ...
5
votes
2answers
355 views

Showing there is only one isomorphism between well ordered sets using transfinite induction

I need to show specifically using transfinite induction that given two well-ordered sets $\left(A,<_{1}\right)$ and $\left(B,<_{2}\right)$ there is only one isomorphism between them. To do ...
4
votes
2answers
151 views

Two questions regarding Ordinal Numbers.

I'm trying to prove that there is an uncountable ordinal all which members are countable ordinal. This is fairly easy if I can state that the class of all countable ordinals is a set and then take the ...
0
votes
1answer
89 views

$\alpha = \omega^\beta(\gamma+1)$

This paper mentions that any ordinal $\alpha < \epsilon_0$ can be written uniquely as $\alpha = \omega^\beta(\gamma+1)$, where $\beta<\alpha$. Does this presentation have a name? Also, how ...
0
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1answer
203 views

Proving of existence of limit ordinal

How to prove that for every ordinal $\alpha$ there exist limit ordinal $\beta$ ,such that$\alpha\in\beta$ ? Thank you.
0
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1answer
104 views

Proving that an initial infinite ordinal is a limit ordinal

As in the title, An initial infinite ordinal is a limit ordinal. How can I start proving that?
8
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2answers
190 views

Non-measurable subset of $\omega_1$

Consider $\omega_1$ equipped with the order topology. Then Borel subsets of $\omega_1$ are precisely those which contain a closed and unbounded set or the complement contains such a set. There must be ...
6
votes
2answers
225 views

Ordinals definable over $L_\kappa$

Suppose $\kappa$ is an uncountable cardinal, with $L_\kappa$ an admissible set (i.e. a model of Kripke–Platek set theory). Let $<_\gamma \subseteq \kappa \times \kappa$ denote a wellordering of $\...
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2answers
341 views

What is the cardinality of the countable ordinals?

Every countable ordinal $\alpha$ can be written uniquely in Cantor canonical form as a finite arithmetical expression, say $C(\alpha)$. We thus have the 1-1 correspondence between the countable ...
3
votes
1answer
117 views

Question about a proof about singular cardinals

The following is a lemma in Just/Weese on page 179: Lemma 17: Let $\kappa$ be an infinite cardinal. Then $\kappa$ is singular iff there exist an $\alpha < \kappa$ and a set of cardinals $\{\kappa_\...
5
votes
2answers
119 views

Question about proof of Hessenberg: $\kappa \cdot \lambda = \lambda$

The following is a theorem: (Hessenberg) Let $1 \le \kappa \le \lambda$ where $\lambda$ is an infinite cardinal. Then $\kappa \cdot \lambda = \lambda$. The proof in the book proceeds by transfinite ...
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vote
1answer
79 views

Constructing a bijection between $\xi$ and $\xi + 1$

I did the following exercise, can you tell me if I have it right, thank you (Just/Weese p 176): Show that $|\xi + 1|$ is either finite or equal to $|\xi|$. (here $\xi$ is an ordinal) By ...
2
votes
1answer
150 views

Proving properties of enumeration of infinite cardinals

I am doing the following exercise from Just/Weese: where $F$ is defined as follows: (a) I'm not sure whether the following passes as "convince myself": Apparently $F$ is defined for all ...
8
votes
1answer
280 views

If $\alpha$ is an indecomposable ordinal, why $\Gamma(\alpha\times{\alpha})=\alpha$?

Greets This is from exercise 3.4 of Thomas Jech's "Set Theory", stated: "Show that $\Gamma(\alpha\times{\alpha})\leq{\omega^{\alpha}}$. Thus $\Gamma(\alpha\times{\alpha})=\alpha$ for all ...
3
votes
2answers
97 views

Ordered Ordinals - continuum hypothesis

I encountered the following problem: If you sort sets by the following: $$A\lt B:\Leftrightarrow \exists f:A\rightarrow B\; injective,\quad \nexists g:B\rightarrow A \; injective$$ You receive the ...
2
votes
1answer
250 views

Does every infinite well-ordered set have an initial segment isomorphic to $\mathbb N$?

This is an exercise question from Chapter 2 of A Course in Galois Theory by D.J.H. Garling: Suppose that $(A,\leq)$ is an infinite well-ordered set. Show that there is a unique element $a$ such that $...
1
vote
1answer
130 views

Proof that for every $X$ $\exists !$ one initial ordinal

The following is an exercise in Just/Weese: Prove that for every set $X$ there exists exactly one initial ordinal $\kappa$ such that $X \approx \kappa$. An ordinal is called initial ordinal if it is ...
0
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0answers
77 views

Problem 1.45 (Boolean-valued ordinals) (J.Bell, boolean-valued models and indipendence proof, 3rd edition)

Problem 1.45 (Boolean-valued ordinals) (J.Bell, boolean-valued models and indipendence proof, 3rd edition) (i) Show that, for any formula φ(x), $ [[ ∃α .φ(α)]] = \bigvee_α [[φ( \hat{α})]]$ and $[[∀α....
2
votes
1answer
105 views

Existence of a $\sup$ and local compactness in the countable ordinals $\Omega$

I understand how the set of countable ordinals $\Omega$ (with the order topology) is not compact, but how is it locally compact? Also, how can the existence of a least upper bound ($\sup$) for a non-...
3
votes
1answer
44 views

Proof of every cofinal subclass of $\mathbf{ON}$ is proper

Can you please tell me if my proof of the following claim is correct? Thank you! Claim: Every cofinal subclass of $\mathbf{ON}$ is proper. Proof: Let $A \subset \mathbf{ON}$ be cofinal. Assume $A$ ...
2
votes
1answer
109 views

Proof that isomorphic strict partial orders have same Mostowski collapse

I tried to prove the following claim, can you tell me if my proof is correct, please? Thank you! Claim: If $\langle X, \prec_X \rangle$ and $\langle Y, \prec_Y \rangle$ are isomorphic strict partial ...
2
votes
1answer
126 views

Proof of “ordinal if and only if is Mostowski collapse”

I tried to prove the following, can you tell me please if my proof is correct? Thank you. Claim: A set $\alpha$ is an ordinal iff $\alpha$ is the Mostowski collapse of a strict well-order $\langle X, ...
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vote
2answers
96 views

Is $\omega_{\alpha}$ sequentially compact?

For an ordinal $\alpha \geq 2$, let $\omega_{\alpha}$ be as defined here. It is easy to show that $\omega_{\alpha}$ is limit point compact, but is it sequentially compact?
5
votes
1answer
118 views

Lower Bound of countable order-types of linearly ordered sets

On Page 44, Set theory, Jech(2006) (Show that) there are at least $\mathfrak{c}$ countable order-types of linearly ordered sets. [For every sequence $a = \{ a_n : n \in N\}$ of natural numbers ...
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vote
1answer
277 views

How to show the following equality of ordinal exponentiation: $n^{\omega^\omega}=\omega^{\omega^\omega}$?

Let $n > 1$ be a positive integer. Prove that $n^{\omega^\omega}=\omega^{\omega^\omega}$ I was able to prove that, since $ω$ is a limit ordinal, and $n > 1$, $$n^\omega= \bigcup_{\beta<\...
1
vote
1answer
260 views

Decreasing sequence of ordinals must stabilize

If $x_1, x_2, x_3, \ldots$ are ordinals such that $x_1\ge x_2\ge x_3\ge \ldots$, then there exists a positive integer $n$ such that $x_n=x_{n+1}=x_{n+2}=\ldots$. I think that this statement is true, ...
3
votes
1answer
86 views

The product of the finite non-zero ordinals, and the product of the finite non-zero cardinals.

I am trying to study for a test I have on Set Theory, and after being given the practice test, I am having some big problems with simple things. One question I am having a lot of problems with is: ...
9
votes
3answers
816 views

Curious facts about ordinal numbers

I have some notes about curious facts about ordinal numbers, for example that their addition is not commutative, multiplication is not distributive from the right hand side and that the exponent rule ...
3
votes
1answer
621 views

Ordinal non-commutative addition example

I know this is a newbie question, so please bare with me :) I'd like to prove within ZF axiomatic set theory that the addition of two ordinals is not commutative. In particular, I'd like to prove ...