BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Kameryn Williams (Bard College at Simon's Rock) DTSTART:20231005T180000Z DTEND:20231005T190000Z DTSTAMP:20260423T021310Z UID:OLS/128 DESCRIPTION:Title: In terpretations and bi-interpretations in second-order arithmetic\nby Ka meryn Williams (Bard College at Simon's Rock) as part of Online logic semi nar\n\n\nAbstract\nThe property of tightness\, introduced by Visser\, give s a notion of semantic completeness for a theory. Specifically\, a theory T is tight if any two distinct extensions of T cannot be bi-interpretable. Important foundational theories like PA and ZF are tight. Consequently in terpretations of extensions of these theories must lose information. For e xample\, ZF + ¬AC can interpret ZFC by restricting to the constructible u niverse while ZFC can interpret ZF + ¬AC via\, essentially\, forcing. But these interpretations destroy information about the original universe\, a nd the tightness of ZF implies there are no alternative interpretations wh ich avoid this problem.\n\nEnayat asked whether the full strength of theor ies like ZF or full second-order arithmetic is necessary for the tightness results and conjectured that this property can be used to give a characte rization of these theories. Phrased in the contrapositive: must it be that any strict subtheory of these theories admits distinct\, bi-interpretable extensions? Alfredo Roque Freire and I investigated this question for sub systems of second-order arithmetic\, providing some evidence for Enayat’ s conjecture. We showed that if you restrict the comprehension axiom to fo rmulae of a bounded complexity then you can find two distinct yet bi-inter pretable extensions of the theory. The main idea of the construction\, not uncommon for work in logic\, goes back to an old observation by Mostowski . Namely\, while truth is not arithmetically definable\, it is definable o ver the arithmetical sets.\n LOCATION:https://researchseminars.org/talk/OLS/128/ END:VEVENT END:VCALENDAR