# 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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### Confusion regarding the requirements for a recursive ordinal notation

According to Wikipedia, an ordinal notation is a function that maps a subset of ordinal encodings to a subset of ordinals. It then mentions Gödel numbering which maps the set of well-formed formulae ...
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### Higher-order equivalence of ordinals

I wonder which ordinals are second-order equivalent, and similarly for other logical equivalences. Let the signature be fixed and include only <. For concreteness, let us first ask for the first ...
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### Reference for tree of bad sequences of WPO

I'm looking for a reference to give in Wikipedia for the following result: Let $X$ be a WPO. Let $T_X$ be the tree of bad sequnces of $X$, and let $o(X)$ be the ordinal height of the root of $T_X$. ...
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### Existence of a almost increase $\omega_1^{\omega_1}$ sequence mod $[\omega_1]^{<\omega_1}$ with length $\omega_2$

In my textbook, the author said that the sequence below is satisfied the requirement. \text{For }\alpha<\omega_1,\forall\gamma<\omega_1,g_\alpha(\gamma)=\alpha, \text{For }\omega_1\le\alpha<...
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### Properties of natural sum and product of ordinals

There seems to be no publicly available listing of properties of natural (or Hessenberg) addition and multiplication of ordinals. So I'm trying to make one. Please confirm correctness of the following ...
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### Is there a canonical mapping between countable transfinite ordinals and $\omega$? What about recursive ordinals?

Consider $\omega^2$. We can build a simple bijection between the ordinal and $\omega$ similarly to how the bijection between $\mathbb{Q}$ and $\mathbb{N}$ can be built. I was wondering if there is a ...
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### Is this extension of n-th derivatives to ordinal-indexed derivatives trivial? [duplicate]

Let $f$ be a function defined everywhere on the real line, which is infinitely differentiable everywhere, in other words, $f$ is everywhere smooth. I define the $\omega$-th derivative, where $\omega$ ...
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### In the constructive theory of direct categories, is it decidable whether an arbitrary morphism is an identity or not?

I'm wondering what the legit definition of direct categories should be in constructive mathematics. I must admit I don't even know in what literature I should look for the definition. I would ...
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### Embedding large countable ordinals into the complex plane

Consider large countable ordinals (e.g. $\epsilon_0$ which is not "large", but still interesting). These are countable sets, so they inject into the complex plane ( or even the real line). ...
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### Does negative trichotomy hold for constructive ordinals?

I don't know if there is a standard term for this, but by "negative trichotomy" I mean ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐴 = 𝐵 ∧ ¬ 𝐵 < 𝐴). This holds for constructive real numbers as an easy ...
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### A few questions about Tychonoff plank

In the Morita's following article (K. Morita. Some properties of M-spaces), constructing an space $X$ and defining an identification on it. My first question is how to prove that $S$ is countably ...
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### Parameter-free $\alpha$-recursivity versus weakened Turing machines with oracles

In the quest to find a transfinite extension of recursivity which matches intuition, mentioned also in my previous question, Discord user onion5 came up with an idea that expressed precisely how I ...
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### Simple $(\alpha+1)$-recursive well-orders with order type $|\alpha\text{-recursive}|$

In the following, $L_\alpha$ is the $\alpha$-th level of the constructible hierarchy, $\alpha$-recursive means definable in $L_\alpha$ by a $\Delta_1$ formula. $|\alpha\text{-recursive}|$ is the ...
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### Iterated $\Pi^1_1$-reflection and non-Gandiness underrepresented in ordinal analyses?

This is a copy of Math StackExchange question #4395977, which I felt was more appropriate for MathOverflow. Note on terminology: "admissible", "$(^+)$-stable", and "$\Pi^1_1$-...
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### How closely do ordinal collapsing functions relate to Skolem hulls?

Ordinal collapsing functions appear in proof theory, and they are used to name large countable ordinals by using uncountable ordinals. Previously I posted a question about why $\psi(\alpha)$ is ...
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### Parameter-free effective cardinals

In the paper "Effective cardinals and determinacy in third order arithmetic" by Juan Aguilera, effective cardinals is defined. I'm curious about its little variation, parameter-free ...
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### Is stable ordinals in non-well-founded model the same as well founded models?

Let $BST$ be the axiom system $KP$ - $\Delta_0$ collection. For an ordinal $\alpha$, we say that $\alpha$ is $\varphi$-$\Sigma_n$-stable, if there is a $\beta>\alpha$ satisfies the formula $φ$ such ...
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### Proof of Theorem Concerning Conway's "Nim Field"

I have a question about the proof of theorem 44 in "On Numbers and Games" on page 58, concerning the "Nim field" ${ON}_2$. As background, ${ON}_2$ turns the ordinals into a field ...
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### Which one of the following two ordinals is larger?

We say that $\alpha$ is $\Sigma_n$-extendable (to $\beta$), if there is $\beta>\alpha$ such that $L_\alpha$ is a $\Sigma_n$ elementary submodel of $L_\beta$. First ordinal: the least $\alpha_0$ ...
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### Gaps in the ordinals writable by Ordinal Turing Machines with a single countable parameter

Let $W(\alpha)$ denote the set of all (countable) ordinals writable by Ordinal Turing Machines with a single (countable) parameter $\alpha$, i.e. each computation starts with a single ($\alpha$-th) ...
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### Why do we need "canonical" well orders?

(I asked this question on Math.SE earlier but received no response and am therefore moving it here, please note that I realise this question is probably incredibly naïve for the experienced set-...
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### The supremum of ordinals eventually writable by Ordinal Turing Machines with an oracle for the class of stabilization ordinals

This question is based on the assumption that all computations start with no ordinal parameters (i.e. the input is empty). The term “stabilization time of a machine” for this question implies the ...
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### Existence of a particular function that maps an arbitrary set of ordinals to a single ordinal

Does there exist a function $f$ that satisfies all of the following three properties? The function converts an arbitrarily large (empty, finite, countably/uncountably infinite) set of ordinals to a ...
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### How closely do ordinal collapsing functions relate to Mostowski collapse?

Ordinal collapsing functions (such as Rathjen's $\psi_\pi$-functions, not the Levy collapse function) name large countable ordinals by mapping larger ordinals below some "large" ordinal, ...
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### Where was the Cantor normal form theorem first proved?

We all take for granted the theorem that every ordinal $\alpha > 0$ has a Cantor normal form, and there are plenty of proofs of it, some of which are on this site. However, where was it proved? Was ...
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### Diagonalization over a normal function and its derivatives on transfinite ordinals

Let $\Phi(0,\beta)$ a normal function from $On$ to $On$, and let $\Phi(\alpha,\beta)$ be the $\alpha$-th derivative of $\Phi(0,\beta)$. For example, let $\Phi(0,\beta)=\aleph_\beta$. Then, all ...
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### Is there a proof of strong normalisation that uses ordinal numbers?

I am currently trying to find a proof for strong normalisation of an extension of $\lambda$-calculus. I've tried several approaches and one would be to assign an ordinal number $\operatorname{cs}(t)$ ...
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### Is this recursion theoretic analogue of a criterion of weakly compact cardinal accurate?

Jensen proved that, if V=L, and $\kappa$ is a regular cardinal, then if for any stationary $A\subseteq \kappa$, the set $\{\alpha\mid A \text{ is stationary below }\alpha\}$ is stationary in $\kappa$, ...
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### When does $\Pi_2$-reflection on $X$ fail to imply iterated $\Pi_1$-reflection on $X$?

Let lowercase Greek letters denote ordinals. Recall from Richter and Aczel's "Inductive definitions and reflecting properties of admissible ordinals", for a set of formulae $\Gamma$ and a ...
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### Can the essence of the $0^\#$ LCA be weakened to an axiom not requiring uncountable cardinals?

"$0^\#$ exists" is an informally stated large cardinal axiom that is to be understood as "there is an uncountable set of Silver indiscernibles", "every uncountable cardinal is ...
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### How could we define "recursively greatly Mahlos"?

A common action in set theory is making a large cardinal axiom "recursive", i.e. turning it from a large uncountable cardinal to a large countable ordinal. For example: Recursively regular =...
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### How to define BHO alternatives below admissible ordinals?

Bachmann-Howard ordinal is a recursive ordinal. It's not that large compared to those proof-theoretic ordinals of stronger theories, but the definition of BHO is sufficient to illustrate how ...
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### If $L_\alpha \vDash ZFC$, then do we have $L_{\alpha+1} \vDash \alpha\text{ is inaccessible}$?

Here we choose the definition of "is a cardinal" as there is no surjective map from a smaller ordinal to it. It's easy to prove that, if $L_{\alpha+1} \vDash\ \alpha\text{ is inaccessible}$, ...
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For ordinals $\alpha<\beta$, we say $\alpha<_{el}\beta$, if there is an elementary embedding with domain $L_\beta$ and critical point $\alpha$. Is $<_{el}$ transitive?