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Results tagged with ordinal-numbers
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user 8991
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.
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 …
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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 …
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15
votes
Consistency of Analysis (second order arithmetic)
As Noah says, the direct successor of Gentzen's method, cut-elimination, has been generalized up to $\Pi^1_2$-comprehension. This was shown separately by Rathjen and Arai; the full results have never …
- 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 …
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