BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Caleb Camrud (Iowa State University) DTSTART:20220113T190000Z DTEND:20220113T200000Z DTSTAMP:20260423T035740Z UID:OLS/82 DESCRIPTION:Title: Con tinuous Logic\, Diagrams\, and Truth Values for Computable Presentations\nby Caleb Camrud (Iowa State University) as part of Online logic semina r\n\n\nAbstract\nGoldbring\,McNicholl\, and I investigated the arithmetic and hyperarithmetic degrees of the finitary and computable infinitary diag rams of continuous logic for computably presented metric structures. As th e truth value of a sentence of continuous logic may be any real in [0\,1]\ , we introduced two kinds of diagrams at each level: the closed diagram\, which encapsulates weak inequalities of truth values\, and the open diagra m\, which encapsulates strict inequalities. We showed that\, for any compu tably presented metric structure and any computable ordinal $\\alpha$\, th e closed and open $\\Sigma^c_\\alpha$ diagrams are $\\Pi^0_{\\alpha+1}$ an d $\\Sigma^0_\\alpha$\, respectively\, and that the closed and open $\\Pi^ c_\\alpha$ diagrams are $\\Pi^0_\\alpha$ and $\\Sigma^0_{\\alpha+1}$.\n\nP roving the optimality of these bounds\, however\, was non-trivial. Since t he standard presentation of [0\,1] with the Euclidean metric is computably compact\, we were forced to work on the natural numbers with the discrete metric (in some sense\, the "simplest" non-compact metric space). Along t he way\, we also proved some surprising combinatorial results. McNicholl a nd I then continued our study of computable infinitary continuous logic an d found that for any nonzero computable ordinal $\\alpha$ and any right $\ \Pi^0_\\alpha$ (or $\\Sigma^0_\\alpha$) real number\, there is a $\\Pi^c_\ \alpha$ (or $\\Sigma^c_\\alpha$) sentence which is universally interpreted as that value.\n LOCATION:https://researchseminars.org/talk/OLS/82/ END:VEVENT END:VCALENDAR