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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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 ...
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387 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 ...
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230 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 < ... < ...
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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 ...
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521 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 $\langle \alpha \times \beta, AntiLex\rangle$ ...
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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 ...
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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 ...
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349 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 ...
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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 ...
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88 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 ...
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200 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.
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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?
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189 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 ...
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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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339 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 ...
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116 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 ...
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118 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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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 ...
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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 ...
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279 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 ...
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96 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 ...
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1answer
249 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 ...
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1answer
129 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 ...
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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 ...
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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 ...
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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 $A$ ...
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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 ...
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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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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?
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117 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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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= ...
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256 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, ...
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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: ...
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809 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 ...
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1answer
614 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 ...
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1answer
140 views

Question about cofinality of an ordinal

Let $\alpha$ an ordinal and $\langle\alpha_\xi\rangle$ a cofinal sequence of elements of $\alpha$. The length, $\gamma$, of this sequence is at least $\operatorname{cf}\alpha$ but can be equal to any ...
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1answer
91 views

Equivalent definition of a closed set.

I try to prove the equivalence between "C is closed in an ordinal $\alpha$" and "each strictly increasing sequence of elements of C of length $< cf\alpha$ converge in C". For $\Rightarrow$, it's ...
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1answer
119 views

An infinite cardinal agrees with all its well-orders on sets of full size.

Suppose $\kappa$ is an infinite von Neumann cardinal (well ordered by $\in$), and take ${<}$ a well-order on $\kappa$. Does there necessarily exists a subset $X\subset\kappa$ of full size (in ...
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4answers
794 views

What's the definition of $\omega$?

This is a follow up on a comment to one of my previous questions. What's the definition of $\omega$? Are the following equivalent definition of $\omega$: $\omega$ is the initial ordinal of ...
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2answers
170 views

Possible inaccuracy in Wikipedia article about initial ordinals

I quote from the Wikipedia article: "So (assuming the axiom of choice) we identify $\omega_\alpha$ with $\aleph_\alpha$, except that the notation $\aleph_\alpha$ is used for writing cardinals, and ...
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103 views

What is a notation for the minimal ordinal of $\mathbb{R}$?

What is a notation for the minimal ordinal of $\mathbb{R}$? I know that $\beth_1$ and $\mathfrak{c}$ designate the cardinality of $\mathbb{R}$, and that $\Omega$ denotes the smallest uncountable ...
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292 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 ...
5
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1answer
173 views

What is the definition of the well-founded part of a model of set theory?

I've been trying to understand John Steel's various notes on inner model theory, but the one thing that trips me up is what he calls the well-founded part of a model of set theory. What exactly is the ...
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3answers
325 views

Other ways of proving that the set of all countable ordinals is uncountable

I know that the standard way of proving that the set of all countable ordinals is uncountable is by stating that if the set is countable, then it incurs Burali-Forti paradox. Is there other ways of ...
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191 views

Question on the use of a parametric version of Transfinite Recursion Theorem in Introduction to Set Theory 3rd ed. by Hrbacek and Jech

My question concerns a proof given on page 118 in the text Introduction to Set Theory 3rd ed. by Hrbacek and Jech. The authors on page 117 prove a version of the transfinite recursion theorem ...
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1answer
381 views

Ordinals with uncountable cofinality

How to construct an ordinal with uncountable cofinality? All the very "large" ordinals I can think of, such as $\omega_\omega^{\omega_\omega}$, still seem to have countable cofinality. I need a better ...
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142 views

Equivalent statement of transfinite/ordinal recursion

I am trying to prove that the "standard" statement of transfinite/ordinal recursion: "Suppose $G$ is a definite operation on partial functions on ordinals. Then there is a unique definite operation ...
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519 views

Which set is unwell-orderable?

In textbook it says that every well-orderable set is equipotent to an initial ordinal number. However, of course unwell-orderable cannot equipotent to any ordinal number, but is there any set is ...
4
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3answers
290 views

A set that is not an ordinal

According to what I heard of, an ordinal is constructed by taking an union of {$\alpha$} $\cup$ $\alpha$ where $\alpha$ is a predecessor ordinal. If so, how can there be a set that is not an ...
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114 views

An inner model and a model that contain same ordinals

In mathematical logic, suppose T is a theory in the language $L = \langle \in \rangle$ of set theory. If $M$ is a model of $L$ describing a set theory and N is a class of M such that $ ...