# Questions tagged [ordinal-numbers]

An ordinal is the order type of a well-ordered set. The first few ordinals are $0, 1, 2, \dots, \omega, \omega+1, \dots$ where $\omega$ is the order type of $\mathbb{N}$, and $\omega+1$ is the order type of $\mathbb{N}$ together with a maximum element.

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### Veblen function with uncountable ordinals & beyond

Disclaimer: I am not a professional mathematician.
Background: I have been researching large countable ordinals for awhile & I think the Veblen function is particularly eloquent. My understanding ...

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### Does every cofinal branch through Kleene's O compute true arithmetic?

My question concerns cofinal branches through Kleene's $O$, which is a set of natural numbers and a computably enumerable relation $<_O$ on this set that provides
ordinal denotations for any ...

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### From a constructive perspective, what are the ordinal numbers?

From a constructive and computational perspective, what are the ordinal numbers?
On the one hand, it seems you can represent ordinal numbers symbolically using something like Cantor Normal Form ...

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### Has the exponentiation of ordinals a nice geometric model?

It is well known that the sum $\alpha+\beta$ of two ordinals $\alpha,\beta$ can be defined "geometrically" as the order type of the sum $(\{0\}\times \alpha)\cup(\{1\}\times\beta)$ endowed with the ...

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### Is Axiom of Choice equivalent to its version for families of sets, indexed by ordinals? [duplicate]

Is Axiom of Choice equivalent to the following statement?
Axiom of Ordinal Choice: For any ordinal $\lambda$ and any indexed family of sets $(X_\alpha)_{\alpha\in\lambda}$ there exists a function $...

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### Infinite positions in 3D chomp

I've recently come back to investigating ordinal chomp. See A winning move for the first player in $3 \times 3 \times \omega$ Ordinal Chomp for a definition. I made a new discovery, that the position \...

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### Transfinite algorithms

The Ford-Fulkerson algorithm is a classic algorithm that computes the maximum flow in a network. It is well-known that if irrational arc capacities are allowed, the algorithm does not necessarily ...

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### What's the use of countable ordinals? (prompted by a remark of Tim Gowers)

In a typically lucid and helpful page of notes for students, A beginner’s guide to countable ordinals, Tim Gowers explains how the countable ordinals can be “constructed rigorously in a way that ...

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### How far does this restricted definition on $\mathcal{O}$ goes?

$\mathcal{O}$ notation describes an onto function $f:\mathcal{O} \rightarrow \omega_{CK}$. In calculating all values $n \in \mathbb{N}$ such that $f(n)=\alpha$, when $\alpha$ is a limit, all indexes $...

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### $Π_2$ strength of KP

I am looking for a characterization of the $Π_2$ statements provable in KP.
Here, KP (often denoted KPω) is the Kripke-Platek set theory, including infinity and full induction on ordinals. Here is ...

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### What is the bijection from $\omega_1^{CK}$ to the natural numbers?

A while back, I learned about the Church-Kleene ordinal and I wondered, since we know that it is countable, what is the bijection from $\omega_1^{CK}$ to $\mathbb{N}$?

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### Can I find the definition of Jager's ordinal collapsing functions?

http://cantorsattic.info/index.php?title=J%C3%A4ger%27s_collapsing_functions_and_%CF%81-inaccessible_ordinals&action=edit
Sadly, Cantor's attic is making an error. This is all I know about Jager'...

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### Examples of proofs using induction or recursion on a big recursive ordinal

There are many proofs use induction or recursion on $\omega$, or on an arbitary (may be uncountable) ordinal. Are there some good examples of proofs which use a big but computable ordinal?
The ...

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### Smallest ordinal modelling $\aleph_1$?

Let $X_1$ be the class of all ordinals $\alpha$ such that there exists a transitive model $M$ of ZF(C) such that $M$ thinks that $\alpha$ is $\aleph_1$.
Every class of ordinals has a minimum element (...

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### What is proof-theoretic ordinal of weak first-order arithmetic?

According to Wikipedia(https://en.wikipedia.org/wiki/Ordinal_analysis) and nlab(https://ncatlab.org/nlab/show/ordinal+analysis), a proof-theoretic ordinal of $\mathsf{PRA}$ is $\omega^\omega$.
...

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### Club filter basis in $\omega_1$

My question is about existing of basis of club filter club($\omega_1$) with cardinality $c$. Does it exist?

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### Which ordinal is larger, the supremum of ordinals writable by iterated Infinite Time Turing Machines or the smallest $\Sigma_2^1$-reﬂecting ordinal?

The ordinal $\tau_1$ corresponds to $\lambda^{\textit{it}}$ (the supremum of all ordinals writable by iterated ITTMs) — see Deﬁnition 3.1 in the paper “ITTMs with Feedback” [Robert S. Lubarsky]. ...

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### Is it an open problem whether fast-growing hierarchies can be defined without fundamental sequences?

Googleology Wiki says this, concerning the relation between fast-growing hierarchies defined for all countable ordinals, and the existence of a system of assigning a canonical fundamental sequence to ...

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### Relation between $\eta$ and $\omega^L_1$

I posted this question on MSE (link: Eventual Writability (general)) about 10 days ago. The current version of this question is a highly abridged version of the one posted there. Let's write "...

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### Why do ordinal collapsing functions use regular cardinals?

Inaccessible cardinals are defined as regular strong limit cardinal, and weakly inaccessible cardinals as regular weak limit cardinal. These cardinals are used by some ordinal collapsing functions. My ...

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### Weaker versions of Gandy ordinals

Gostanian's paper "The next admissible ordinal" (see https://www.sciencedirect.com/science/article/pii/0003484379900251 ), is concerned with the supremum of the $\alpha$-recursive ordinals for various ...

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### Compact definition of ordinals

This question is about von Neumann's informal definition of ordinals as "sets of all smaller ordinals" and was discussed in this math.stackexchange question.
When trying to formalize this definition, ...

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### Computable models of the ordinal numbers

It's known, for example in the answer to this question: Is there a computable model of ZFC? that ZFC has no computable model. My questions is: is there a model of ZFC for which the order relation on ...

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### Generalizing König's Lemma

In some recent work, I need a strengthening of König's Lemma to "trees" of arbitrary ordinal heights. Trees, in this context, are really just well-founded partially ordered sets. See, for instance, ...

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### Capturing the $\omega_1^{\mathrm{CK}}$-th stage of Gödel's constructible hierarchy

For an ordinal $\alpha$, let $L_\alpha$ be the $\alpha^{th}$ set of Gödel's constructible hierarchy and let $\omega_1^{\mathrm{CK}}$ be the first non-recursive ordinal or the first admissible ordinal ...

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### Is the least ordinal containing all countable ordinals defined by a formula an element of itself? (Out of date, see below)

[UPDATE] The question has been updated and is mostly unrelated to the question above the line. The updated question is below the line.
The following argument supports a "yes" answer; is it convincing?...

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### Coloring triples in trees

Definitions
Let us say a tree is a partially ordered set $(P, \leqslant )$ such that for any $t\in P$, the ancestor set $\{s\in P: s\leqslant t\}$ is finite and linearly ordered. We let $MAX(P)$ ...

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### Function on the set of limit countable ordinals

Let $\Lambda$ be the set of all countable limit ordinals. Does there exist an injective function $f:\Lambda\to\omega_1$ with the properties:
$\forall \lambda\in\Lambda:~f(\lambda)<\lambda$
$\...

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### Simple transfinite generalization of $p$-adic integers

One way to define the ring of $p$-adic integers is as a quotient of the formal power series semiring $\Bbb N[[x]]/(x-p)$. One can likewise start with the formal power series ring $\Bbb Z[[x]]/(x-p)$ ...

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### Countable union of well ordered sets [closed]

Assume I have a sequence $(A_i)_{i<\omega}$ of well-ordered subsets of an ordered set $S$. Assume that $A:=\underset{i<\omega}{\cup}A_i$ is also well-ordered. Let $\alpha$ be an ordinal upper ...

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### An Apparent Incongruity

I thought of a certain point that seems to point to an apparent incongruity. Hopefully someone would promptly point out the logical mistake and/or some implicit assumption that may not hold under a ...

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### Partial well-ordering of formulas

Given a theory $T$, for arbitrary formulas $φ$ and $ψ$ that provably in $T$ denote an ordinal, set $[φ]_T < [ψ]_T$ iff provably in $T$, the ordinal denoted by $φ$ is less than the ordinal denoted ...

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### Hartogs' Number of the Reals and $\Theta$ without choice

There are two important numbers that in some meaningful sense describe "how well-orderable" the reals are:
Hartogs' Number $H(\Bbb R)$, also notated as $\aleph(\Bbb R)$, the least ordinal/well-...

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### Is it possible to construct a formal language that allows to refer to specific real numbers that encode ordinals accidentally writable by an ITTM?

Let $A$ denote a particular (fixed) algorithm to encode ordinals as real numbers. The exact technical description of $A$ is irrelevant for this question: it can be any algorithm that is mathematically ...

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### Natural examples of recursive pseudowellorderings

Question: What are some natural examples of recursive pseudowellorderings?
By natural, I mean in the style of reasonable ordinal notation systems as opposed to dependent on a Gödel numbering or an ...

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### Ordinal analysis and nonrecursive ordinals

Ordinal analysis is typically described as characterizing recursive ordinals in a theory $T$, but there is a sense in which it can characterize all $T$-ordinals, even those that are nonrecursive.
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### Relation of $\omega_{\omega_1+1}^{CK}$ to some other ordinals

This was posted as a side question in Formal definition of this ordinal? and was split as a separate question based upon suggestion in comments there.
Assume an ordinary ORM model (call it $C_1$). ...

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### Formal definition of this ordinal?

Few days I asked this question (https://math.stackexchange.com/questions/2907733/simple-ordinal-question) on MSE. Summary of the question is that I defined a certain function over ordinals $x \mapsto \...

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### Which finite sets could be packed into a square?

This question is inspired by an interesting visualization of the finite levels of von Neumann's hierarchy on Adam P. Goucher's blog, Complex Projective 4-Space.
The problem starts with a two-...

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### Is the measurable space $(\omega_1,\mathcal{P}(\omega_1))$ separable?

Here $\omega_1$ is the first uncountable ordinal, and $\mathcal{P}(\omega_1)$ denotes the power set of $\omega_1$. Separable means countably generated as a $\sigma$-algebra.

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### Reference Request: Existence of Ordinal Rank Theory?

Notations: Recall that $\omega_1$ is the first uncountable ordinal.
Let $X$ be a Polish space (completely metrizable and separable) and $F(X)$ be the collection of all real-valued functions on $X.$
...

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### Ordinal vs. cardinal dimension

$\newcommand{\Ord}{\operatorname{Ord}}$When does it make sense to define the dimension of a space to be an infinite ordinal, instead of restricting to infinite cardinals?
We would essentially have to ...

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### $\omega_1$-like elementary chains from long countable elementary chains

For some countable first-order theory $T$, if we have a linearly ordered set $I$ and an elementary chain $\{\mathfrak{A}_i\}_{i\in I}$ we can form a structure $\mathfrak{A}_{I}^\star = (\bigcup_{i\in ...

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### Example of an $\omega_1$ decreasing chain of dense semicontinua?

In his well-known paper Bellamy constructs an indecomposable continua with exactly two composants. The setup is as follows:
We have an inverse-system $\{X(\alpha); f^\alpha_\beta: \beta,\alpha < \...

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### Large cardinals disproving J=L

Given $J_\alpha$ denotes the $1+\alpha$th term in the Jensen Hierarchy and $J=\bigcup_{\alpha\in On}J_\alpha$, are there any known large cardinals $\mathfrak{K}$ such that $\text{ZFC}+\mathfrak{K}$ is ...

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### Ordinal corresponding to well-quasi-order on graphs

Let $K$ be an infinite cardinal. Then, by the Robertson–Seymour theorem, the set of graphs with fewer than $K$ vertices and edges form a well-quasi-order.
In terms of $K$, what is the maximal order ...

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### What is the Proof-Theoretic Ordinal of KPh?

If we work in this notation:
$$C_0 (\alpha, \beta) = \beta \cup \lbrace 0 \rbrace$$
$$C_{n+1} (\alpha, \beta) = \lbrace \gamma + \delta, \omega^\gamma, \Omega_{\gamma}, I_{\gamma}, \psi_\pi(\eta) | \...

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### Ramsey's theorem for the first uncountable ordinal

Sierpiński proved that a version of Ramsey's theorem for colourings of pairs of countable ordinals fails miserably by comparing the ordering of $\omega_1$ with the linear ordering of (a subset of) the ...

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### A question about ordinal analysis

I have several questions related to ordinal analysis.
According to [1], here are the proof-theoretic ordinal of some well-known theories (using $|T|$ do denotate the proof-theoretic ordinal of $T$):
...

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### Mapping Between Notations - 2

As the name implies, this question is somewhat similar in spirit to the previous question I asked with same title. This question is also about existence (or lack thereof) of certain possibilities ...