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Math Logic Seminars
Schedule for Fall 2009 Visit http://logic.cmu.edu/seminar/seminar.html for seminar details.TUESDAY, October 6, 2009William Boney, Carnegie Mellon University Model theory of L_{omega_1, omega} ABSTRACT: L_{omega_1, omega} is the language we get when we start with L and allow for countable conjunctions and disjunctions. Among other things, this allows us to talk about things involving finiteness and countability and allows us to omit countable types with a sentence. However, we lose the Compactness Theorem and the Upward Lowenheim-Skolem-Tarski Theorem, our two favorite methods for constructing models. In this talk, we introduce L_{omega_1, omega} and some of the model construction methods that aim to replace these. Time: 12:00 P.M. TUESDAY, November 3, 2009Peter Lumsdaine, Carnegie Mellon University Fixed-point theorems, constructively ABSTRACT: The Bourbaki-Witt and Tarski fixed-point theorems for chain-complete posets are two minor gems of early set theory: simple and natural statements about posets, easily and elegantly proven with transfinite induction. In constructive logic, the situation is less simple. I will sketch a counterexample (in the effective topos, a realisability model) showing that neither can be proven entirely constructively; and I will show how they are in fact equivalent to the existence of enough ordinals. Time: 12:00 P.M. TUESDAY, December 1, 2009Ed Dean, Carnegie Mellon University Randomness III ABSTRACT: In this final talk of the series, I have two overall goals. First, I will look at some alternatives to Martin-Lof's notion of randomness, and explain the relationships among the different concepts, using martingales as a common thread. Second, I will discuss (in not too fine detail) certain effective versions of classical probabilistic laws, for various of the randomness notions presented. This extends the discussion begun in the previous talk, in which Jason Rute presented the result that the strong law of large numbers holds for every Martin-Lof random sequence. This is the the third in a series of talks by Ed Dean and Jason Rute. Time: 12:00 P.M.
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