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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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
248 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
128 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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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 ...
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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 $A$ ...
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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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1answer
124 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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1answer
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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1answer
272 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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1answer
248 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
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1answer
84 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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3answers
804 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
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1answer
593 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 ...
3
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1answer
139 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 ...
3
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1answer
118 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
766 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 ...
2
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2answers
163 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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2answers
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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2answers
290 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
169 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
300 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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1answer
189 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
373 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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1answer
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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3answers
491 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
289 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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2answers
110 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 $ ...
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4answers
251 views

Which axiom shows that a class of all countable ordinals is a set?

As stated in the title, which axiom in ZF shows that a class of all countable (or any cardinal number) ordinals is a set? Not sure which axiom, that's all.
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278 views

How does one prove that there are uncountable number of countable ordinals? [duplicate]

Possible Duplicate: Uncountability of countable ordinals How does one prove that there are uncountable number of countable ordinals? Obviously, there are equal to or more than countable ...
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1answer
737 views

How to prove that Cantor's normal form can produce all ordinal numbers

How do we prove Cantor's normal form can produce all ordinal numbers? Also, it's a bit difficult to picture $\omega_1$ as a format in Cantor's normal form. Can anyone show how to do this?
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2answers
229 views

How to prove that $\epsilon_0$ exists?

How do we prove that $\epsilon_0$ exists? The definition of $\epsilon_0$ says that $\omega^{\epsilon_0} = \epsilon_0$. So, how do we know whether such number exists?
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How to define countability of $\omega^{\omega}$ and $\omega_1$? in set theory?

How is the ordinal $\omega_1$ defined? I know that it is a supremum of all smaller ordinals, but then $\omega^\omega$ is also a supremum of all smaller ordinals. How can we distinguish these two ...
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1answer
151 views

Generalization of ordinals to well-founded sets?

In set theory, the ordinal numbers are objects that represent the order types of well-ordered sets. Is there a similar class of objects that represent the order types of well-founded sets? If so, ...
3
votes
2answers
251 views

Why would the axiom of choice be needed if ordinals are well-ordered without AC?

Will ordinals be well-ordered without AC? This seems to be obviously true, as they are by definition well-ordered. Why would we then need the axiom of choice. We can just form a bijection function to ...
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3answers
519 views

What does $\upharpoonright$ in $G(F\upharpoonright\alpha)$ mean?

More formally, we can state the Transfinite Recursion Theorem as follows. Given a class function $G\colon V\to V$, there exists a unique transfinite sequence $F\colon\mathrm{Ord}\to V$ (where ...
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1answer
197 views

Limit ordinals vs. Points at infinity

The point at infinity $\infty$ makes $\mathbb{R}$ into a topological circle, i.e. into the real projective space $\mathbb{R}P$: $\infty$ enhances the original structure ($\mathbb{R}$) and makes it ...
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235 views

Question on a proof from Jech's Set Theory of: If $X$ is a set of cardinals, then $\sup X$ is a cardinal.

The proof: Let $\alpha= \sup X$. If $f$ is a bijective mapping of $\alpha$ onto some $\beta < \alpha$, let $\kappa \in X$ be such that $\beta < \kappa \le \alpha$. Then $|\kappa|=|\{f(\xi): \xi ...
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677 views

How to prove that the set of all countable ordinals, $\omega_1$, is uncountable / has the cardinality $\aleph_1$?

The only reasoning I've seen given for this is that it's uncountable because it can't include itself an element. I'm a little unconvinced and was looking for a more proper formal proof demonstrating ...
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1answer
376 views

Discussion: Differing definitions for the rank of a set

I've just identified that the definition we used for the rank of a set in my set-theory class (1.) is different than the one I commonly find on the web (2.). $\text{rank}(A)=\min\{\alpha\mid ...
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4answers
359 views

How do I show there isn't an order isomorphism b/w the two sets $\{1, 2, 3,…\}$ and $\{1, 2, 3, …, \omega \}$

That is, how can I prove there isn't a bijection $f$ from one set to the other such that $f(x) < f(y)$ iff $x < y$?
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2answers
175 views

What is the name of $V_\alpha$?

In the Von Neumann cumulative hierarchy, $V:=\bigcup_\alpha(V_\alpha)$ is called the universe. Is there a name for the individual levels $V_\alpha$? Just as one can say "The closure of $A$ is defined ...
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1answer
250 views

If $\kappa$ is a cardinal, $\aleph$ is any $\aleph$-number, and if $\kappa\leq\aleph$ then $\kappa$ can be well ordered as well.

I'm having trouble understanding the statement: If $\kappa$ is a cardinal, $\aleph$ is any $\aleph$-number, and if $\kappa\leq\aleph$ then $\kappa$ can be well ordered as well. I understand the ...
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1answer
271 views

Does there exist a prime number decomposition for ordinal numbers?

As is well known, each natural number (except $0$) can be written uniquely as product of finitely many prime numbers (with $1$ being the empty product). My question is: Does some analogue theorem also ...
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2answers
280 views

Which algebraic structure captures the ordinal arithmetic?

Consider the set class $\mathrm{Ord}$ of all (finite and infinite) ordinal numbers, equipped with ordinal arithmetic operations: addition, multiplication, and exponentiation. It is closed under these ...
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376 views

How does the supremum of the cardinalities of a set of ordinals relate to the supremum of the ordinals?

Suppose we have a set of ordinals such that their supremum is a cardinal (not in the original set). $\sup\{\alpha\}=\kappa$ I'm interested in the supremum of the cardinalities of those ordinals: ...
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1answer
195 views

When collapsing a cardinal, what ordinal does it become?

Work in $V$. Let $P = \text{Col}(\omega, \omega_1)$ and suppose that $G$ is generic for $P$ over $V$. Then $V[G]\models |\omega_1^V|=\aleph_0$ and $\omega_2^V=\aleph_1$. In particular, ...
9
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287 views

$\varepsilon$-number countability without choice

Let $\alpha\mapsto\varepsilon_\alpha$ be the enumeration of the $\varepsilon$-numbers--that is, those $\alpha$ such that $\omega^\alpha=\alpha$--by the ordinals. If we know that countable unions of ...