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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.
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 …
- 2,031
1
vote
Accepted
Ordinal numbers reachable by primitive recursive ordinal functions in omega
According to Jeremy Avigad's "An ordinal analysis of admissible set theory using recursion on ordinal notations" (corollary 4.2), if $\alpha$ is closed under the primitive recursive ordinal functions, …
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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
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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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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1
vote
How closely do ordinal collapsing functions relate to Mostowski collapse?
This description seems to be a closer connection to Mostowski collapse than "it is the order type of $C_\Omega(\alpha,\rho)$ when some elements are removed":
A similar comparison appears in Toshiyasu …
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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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6
votes
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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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
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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1
vote
What's the order type of the following set?
Edit 2024: I now think that this answer is wrong, specifically that the result in the last paragraph does not answer the original question. I am reluctant to delete this answer as it would also delete …
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0
votes
Transfinite sums related to a sequence
About questions 3 and 4:
It's known that for any countable ordinal $\alpha$ there is a subset of the reals with order type $\alpha$ under the usual order. Using this, given a countable ordinal $\alph …
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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