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Results tagged with ordinal-analysis
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user 8991
8
votes
Accepted
Which ordinals are proof-theoretic ordinals?
If $\alpha$ is a reasonable presentation of a computable ordinal then the proof-theoretic ordinal of $ACA_0+\alpha$ is well-ordered is the smallest $\epsilon$ number $>\alpha$. (There's a proof of th …
- 7,145
22
votes
Accepted
Why do stacked quantifiers in PA correspond to ordinals up to $\epsilon_0$?
8/14: Substantially edited in response to comments: added to 1st part, added new 2nd and 4th parts
There's also some discussion underneath, and a link to a partial write-up of a case of cut-eliminati …
- 7,145
16
votes
Accepted
What is the proof-theoretic ordinal of PA + Con(PA), PA + Con(PA + Con(PA)) etc., and why?
This is an instance of a general phenomenon: adding true $\Pi_1$ sentences to a reasonable theory doesn't change its proof-theoretic ordinal. $Con(T)$ is $\Pi_1$, so if $T$ is any consistent theory, …
- 7,145
11
votes
Accepted
Correspondence between proof-theoretic ordinals and fast growing functions?
Yes: for many theories, $\alpha$ is the proof-theoretic ordinal of T exactly when T proves that $f_\beta$ is total for all $\beta<\alpha$, but does not prove $f_\alpha$ is total. (Where $f_\alpha$ is …
- 7,145
33
votes
Accepted
Is there a computable ordinal encoding the proof strength of ZF? Is it knowable?
Rom Maimon is describing the program of proof-theoretic ordinal analysis.
First, as you observed in your addendum, it isn't interesting to find some encoding of an ordinal whose well-foundedness im …
- 7,145
6
votes
What is the proof-theoretic ordinal of Hyperarithmetical Comprehension?
The difficulty is in what it means to go up the ramified hierarchy. When talking about theories, you can't write down a computable or c.e. theory which perfectly captures the whole ramified hierarchy …
- 7,145
5
votes
What is the proof-theoretic ordinal of Hyperarithmetical Comprehension?
This is a second answer (think of it as part two of the other, already long, answer), in response to the clarification of the question by Keshav Srinivasan in the comments on that question.
First, no …
- 7,145
13
votes
Why is there a need for ordinal analysis?
In addition to the reasons Andreas gives, Gentzen's theorem gives additional information that's interesting even if you don't have any qualms about consistency.
In particular, ordinal analysis gives …
- 7,145
8
votes
Accepted
Models of PRA/EFA with induction on $X$ but not $\omega^X$
I suspect the paper you want is Avigad and Sommer, A Model-Theoretic Approach to Ordinal Analysis. As the name suggests, they give an ordinal analysis rooted in the structure of models of arithmetic. …
- 7,145
13
votes
Accepted
Nontrivial upper bounds on proof-theoretic ordinals of strong theories: do we have any?
I think the short answer is no, and I'm not sure there really could be: I don't think we "know" any ordinals above the ordinal of $\Pi^1_2-CA$ but below $\omega_1^{CK}$. Theoretically someone could w …
- 7,145
9
votes
Which ordinals can be proof-theoretic ordinals of a reasonable theory?
Regarding the small Veblen ordinal, Rathjen and Weiermann gave an analysis of theories in that range of strength in Proof-theoretic investigations of Kruskal's theorem.1 Working over a reasonable bas …
- 7,145