Computational Prospects of Infinity II: AII Graduate Summer School
(15 Jun - 13 Jul 2011)
## ~ Abstracts ~
Chris Conidis, University of Waterloo, CanadaWe will examine the precise definability complexity of the prime radical and Jacobson radical in commutative and noncommutative rings. More specifically, we will show that the Jacobson radical is always \Pi^0_2 definable, while the prime radical is \Sigma^0_1 definable in commutative rings and \Pi^1_1 definable in noncommutative rings.
Grigor Sargsyan, University of California at Los Angeles, USAWe will introduce genericity iterations and use them to show that various degrees of definability can be captured by canonical models of ZFC. We will then explore the MSC. The talk is intended for graduate students.
Frank Stephan, National University of SingaporeThe current work introduces the notion of pdominant sets and studies their recursion-theoretic properties. Here a set A is called pdominant iff there is a partial A-recursive function psi such that for every partial-recursive function phi and almost every x in the domain of phi there is a y in the domain of psi with y leq x and psi(y) > phi(x). While there is a full Pi-0-1-class of nonrecursive sets where no set is pdominant, there is no Pi-0-1-class containing only pdominant sets. No weakly 2-generic set is pdominant while there are pdominant 1-generic sets below K. The halves of Chaitin's Omega are pdominant. There is a high r.e. set which is not pdominant. No set which is low for Martin-Loef random is pdominant. Joint work: This is joint work with Chitat Chong, Gordon Hoi and Dan Turetsky.
Hugh Woodin, University of California at Berkeley, USANational University of Singapore We present the Martin-Steel theory of extender models and prove comparison assuming nice-UBH.
Hugh Woodin, University of California at Berkeley, USANational University of Singapore We show that the Martin-Steel theory cannot be extended past the level of superstrong cardinals.
Hugh Woodin, University of California at Berkeley, USANational University of Singapore We survey the results which show that the problem of building an extender model at the level of one supercompact cardinal is the "last" inner model problem. |
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