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Results tagged with ordinal-numbers
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user 479330
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.
6
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Smallest ordinal modelling $\aleph_1$?
About well-definedness of $X_1$, "being a model of ZFC" is definable since ZFC is a recursive theory, so we could construct some $\Sigma_1^0$ predicate $\textrm{isZFCAxiom}(e)$ for $e$ a Godel-coding …
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6
votes
0
answers
175
views
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 "$\Pi^1_1$-reflecting" can all be fo …
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5
votes
1
answer
297
views
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, often chosen …
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4
votes
0
answers
151
views
Slicing large countable ordinal properties, from $\Pi_3$-reflection to $\Sigma_2$-admissibility
Edit 2024: This post was based on an incorrect premise, as can be seen by my conversation with Farmer S in the comments. However the mistake I made and the conversation in comments may be instructive …
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3
votes
Set theory: fixed points of $n \mapsto \varepsilon_n$ and $n \mapsto \omega_n$
As Noah Schweber mentioned, the Veblen function is relevant for fixed points of $\alpha\mapsto\varepsilon_\alpha$. The Veblen hierarchy starts with $\varphi_0(\alpha)=\omega^\alpha$, and each $\varphi …
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3
votes
What is the proof-theoretic ordinal of KPh?
Let $\mathsf{KPh}$ denote the theory $\mathsf{KP}+``\textrm{The recursively inaccessibles are unbounded}\! "$. I haven't found an explicit analysis of $\mathsf{KPh}$ in Rathjen's preprints, but there …
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3
votes
1
answer
288
views
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 class of ord …
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3
votes
Parameter-free effective cardinals
Edit Jul 25: These results may be strengthenable by using theorem 7.8 of chapter V of Admissible Sets and Structures instead of lemma 1, I may edit this post in the future with any resulting improveme …
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3
votes
Accepted
A question on the size of an admissible ordinal
Some references to literature: In "Reflection and Partition Properties of Admissible Ordinals" (Annals of Math. Logic vol. 22, iss. 3, 1982), Kranakis defines a $\Sigma_n$-admissible ordinal to be an …
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2
votes
Diagonalization over a normal function and its derivatives on transfinite ordinals
No, not all $G_\beta$ are normal. For example let $\Phi(0,\beta)$ be any normal function whose least fixed point is greater than $\omega$ and consider $G_\omega(\alpha)$. Since $\Phi(\beta+1,0)>\omeg …
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2
votes
What's the use of countable ordinals? (prompted by a remark of Tim Gowers)
As mentioned in Prof. Weaver's answer, the graph minor theorem (also known as the Robertson-Seymour theorem) uses large countable ordinals. The graph minor theorem is quite relevant to non-set-theoret …
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2
votes
1
answer
159
views
Ordering patterns of projecta by least witness
Let $J$ denote Jensen's modification of the constructible hierarchy. For an ordinal $\alpha$ and an $n\in\mathbb N^+$, let $\rho_n^{J_\alpha}$ denote the $\Sigma_n$-projectum of $J_\alpha$, the least …
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2
votes
0
answers
185
views
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 someti …
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2
votes
Which ordinals can be proof-theoretic ordinals of a reasonable theory?
$\newcommand{\bomega}{\boldsymbol\omega}$Given the definition of bounding ordinal in the post and the potential sensitivity to coding mentioned in edit 2, these seem to be two main ways to formalize b …
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1
vote
Ordinal notations in α-recursion theory
A concept that appears a lot in the context of ordinal notations in recursion theory is a "copy", which is a set (usually of lower rank) equipped with an ordering that's isomorphic to some other set. …
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