# 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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### 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 ...
215 views

### 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 ...
245 views

### 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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### Is this relation about elementary embedding transitive?

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?
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### Existence of a generalized stable ordinal

An ordinal $\alpha$ is called +1 stable, if $L_\alpha <_1 L_{\alpha+1}$ By considering $Σ_n$ elementary submodel we can generalize it. I'm curious about its further generalizations. Does there ...
123 views

### Ordinal notations in α-recursion theory

Is there a theory about using α-recursion to compute ordinals? For example, consider α-recursive well orders on α, what is the supreme of their order type? Is it the next admissible ordinal after α? ...
287 views

### How large are the stabilization times of Ordinal Turing Machines with an oracle for the transfinite initial ordinals?

This question is based on the assumption that $V \ne L$ and we have $\omega_1^L < \omega_1$ (here $\omega_1^L$ is equal to the supremum of ordinals accidentally writable by no-oracle Ordinal Turing ...
236 views

### What does $\omega^*$ mean? [closed]

I've recently found in some short article (source below) the symbol $\omega^*$ (generally, starred ordinal number), but without explanation what that symbol means. From the context I understood that ...
324 views

### Countably infinite sets of ordinals as parameters for Ordinal Turing Machines

Let $A$ and $B$ denote two countably infinite sets of ordinals. Let $W_A$ denote the supremum of ordinals writable by Ordinal Turing Machines with the set $A$ given as the source of parameters. That ...
186 views

### Is there a real $x$ which is eventually writable from an ordinal parameter $\alpha < \omega_1$, but not from $\omega_1$?

According to Lemma 3.14 in the paper “Recognizable sets and Woodin cardinals: Computation beyond the constructible universe”, there is a real $x$ in $L$ which is recognizable from some ordinal $\alpha$...
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### Is the supremum of L-definable cardinals silver-indiscernible

Let $\kappa$ be the supremum of ordinals first order definable in L without parameters. Assume $0^\sharp$ exists. Is $\kappa$ the least silver indiscernible ordinal?
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### How to compare three supremums of ordinals eventually writable by Ordinal Turing Machines?

This question implies that we have fixed: (i) a particular enumeration of Ordinal Turing machines; (ii) a particular way to encode an ordinal by an infinite binary sequence. The class of $$-...
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### How large is the supremum of halting times of Infinite Time Turing Machines, assuming that halting times are bounded and inputs are arbitrary?

Given a fixed enumeration of Infinite Time Turing Machines (ITTMs), let $M_i(x)$ denote a computation of an $i$-th ITTM, assuming that the input $x$ is a real (an infinite binary sequence). Then the ...
172 views

### Ordinal numbers reachable by primitive recursive ordinal functions in omega

$\def \PRo {{\mathcal { PR } _ \omega}}$ The class of primitive recursive ordinal functions in the constant omega function (henceforth denoted by $\PRo$) are defined by Jensen and Karp (1971) as ...
356 views

### Cardinality of infinite towers of Alephs - can tower be more than countable?

Lets define function T as $$T(0) = \aleph_0$$ $$T(1) = \aleph_{\aleph_0}$$ $$T(2) = \aleph_{\aleph_{\aleph_0}}$$ etc No finite tower of alephs can reach the first inaccessible cardinal My questions ...
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### Definability of Gödel's pairing function on ordinals

Given an infinite cardinal $\kappa$, Gödel's function is a well-known bijection $p:\kappa^2\to\kappa$. Is $p$ definable in the structure $\langle\kappa;\in\rangle$? Is $p$ definable in a bigger 2nd ...
356 views

### What comes after the Bachmann Howard Ordinal?

I've read and known what the FGH is all the way up to a>Bachmann Howard ordinal which is ψ(Ω_2), the Googology Wiki, Wikipedia, Chris Bird articles, and YouTube don't show how the Fast Growing ...
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### Countable Fodor's Lemma?

Does Fodor's lemma fail for countable ordinals? For the usual statement of Fodor's lemma to make sense, one needs well-behaved notions of club and stationary sets, which fail for countable ordinals, ...
1 vote
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### How large is the smallest ordinal larger than any “minimal ordinal parameter” for any pair of an Ordinal Turing Machine and a real?

In this question, the notation $P^x(\alpha)$ denotes a situation where a particular OTM-program $P$ performs a computation on input $x$ with an ordinal parameter $\alpha$, assuming that $x$ is written ...
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### Where can I read about Veblen functions / klammersymbols beyond the large Veblen ordinal?

So, I'm not sure to what extent this is a thing. John Baez mentions in this blog post that common large countable ordinals beyond the large Veblen ordinal can also "be defined as fixed points&...
186 views

### Higher order arithmetic, hierarchies and proof theoretic ordinals

I asked this question on MSE some days ago but I have not received any answer so I have decided to post it here. I would like to consider a generalization of the notation $\Pi$ and $\Sigma$ used for ...
302 views

### Definability of ordinals in various signatures

Recently, I've been studying what the definable subsets of the countable ordinals "look like" from the perspective of bare-bones first order logic (not set theory) equipped with various ways ...
107 views

### Trees of prescribed ordinal

My question is very imprecise, as I know very little about descriptive set theory. In a problem I am thinking about I have a family of well-founded trees (finite sequences on $\cup_n X^n$ closed under ...
643 views

### Intuition about ordinal fixed points $\alpha = \aleph_\alpha$

I wanted to ask for your intuition about ordinal fixed points $\alpha = \aleph_\alpha$, where $\aleph_\alpha$ stands for the $\alpha$-th Aleph number in the Aleph sequence of cardinalities. For ...
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### Self-contained proof of WO of Buchholz's ordinal notation system

I would like a self-contained proof that the ordinal notation system defined by Buchholz in this paper is indeed well-ordered. Meaning, I would like a proof that does not rely on ordinals. Buchholz's ...
277 views

### For which ordinals do we have $V_\alpha = L_\alpha$?

Some elements of $L$ become constructible only in levels higher than its rank level. So I ask: Let $V$ be such that $V = L$. For which ordinals $\alpha$ do we have $V_\alpha = L_\alpha$? Indeed, we ...
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### A connected topological space whose points cannot be connected by irreducible components

Does there exist a topological space $X$ with the following properties? $X$ is connected. The set of irreducible components of $X$ is locally finite. Not every pair of points in $X$ can be "connected ...
487 views

### 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 ...
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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 ...
117 views

### 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 ...
331 views

### 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 (...
268 views

### 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?

Googology 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 ...