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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
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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
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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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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1
vote
How to solve this exercise about large countable ordinals?
This is a partial result, which is that there is an $(M,E)$ satisfying all but the end-extension requirement, in place of $M$ being an end-extension of $L_{\gamma_1}$, there is just an $X\subseteq M$ …
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6
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0
answers
175
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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 "$\Pi^1_1$-reflecting" can all be fo …
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2
votes
0
answers
185
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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 someti …
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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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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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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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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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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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5
votes
1
answer
297
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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, often chosen …
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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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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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0
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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 …