Timeline for Natural $\Pi^1_2$ (or worse) classes of structures?
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Jan 14, 2022 at 17:40 | comment | added | Noah Schweber | Oh thanks, that's great! | |
| Jan 14, 2022 at 17:37 | comment | added | Ulrik Buchholtz | Girard has put Proof theory and logical complexity II on his site here: girard.perso.math.cnrs.fr/Archives4.html | |
| Jan 6, 2015 at 21:54 | vote | accept | Noah Schweber | ||
| Oct 27, 2013 at 23:39 | comment | added | Ulrik Buchholtz | I found Eléments de logique $\Pi^1_n$ after all. But I don't know how to pronounce ptyx/ptykes nor why Girard picked the name. :) BTW, don't bother tracing the second volume of Girard's book, Proof theory and logical complexity; it never appeared (I hear there are drafts out there, but I haven't seen any). | |
| Oct 27, 2013 at 23:09 | comment | added | Noah Schweber | The abstract looks interesting; however, I have to ask: how is one supposed to pronounce "ptyx?" And why "ptyx" in the first place? jstor.org/stable/2907653 | |
| Oct 27, 2013 at 22:37 | comment | added | Ulrik Buchholtz | Indeed, I think this is in this article by Ressayre. See Sec. 3 there (the objects are called ordered Ehrenfeucht-Mostowski models with finitely many function symbols, but I think they're the same thing as (codes of) weakly finite dilators). There are related notions (ptykes) at higher types that provide $\Pi^1_n$-complete notions, according to a hard-to-find paper by Girard and Ressayre. | |
| Oct 27, 2013 at 17:34 | comment | added | Noah Schweber | Thanks! I'd definitely be interested in whether weakly finite dilators are a $\Pi^1_2$-complete set of reals - that certainly seems plausible. | |
| Oct 27, 2013 at 17:18 | comment | added | Ulrik Buchholtz | I added the definitions. Rereading your question, I see you were looking at sets of reals, while my answer addresses subsets of $\omega$. But perhaps weakly finite dilators are a $\Pi^1_2$-complete set of reals? | |
| Oct 27, 2013 at 17:15 | history | edited | Ulrik Buchholtz | CC BY-SA 3.0 |
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| Oct 27, 2013 at 7:09 | comment | added | Noah Schweber | I'm finding Girard alternately vague and difficult to read; what is a "recursive dilator?" My understanding is that a dilator is a functor from ON to ON preserving limits and pullbacks. The obvious (to me) way of recursifying this results in recursive dilators being indexed by natural numbers, which isn't exactly what I was looking for. | |
| Oct 27, 2013 at 1:17 | review | First posts | |||
| Oct 27, 2013 at 1:18 | |||||
| Oct 27, 2013 at 1:00 | history | answered | Ulrik Buchholtz | CC BY-SA 3.0 |