Computational Prospects of Infinity II: AII Graduate Summer School
(15 Jun - 13 Jul 2011)

Jointly funded by the John Templeton Foundation


~ Abstracts ~

 

The definability of radicals in commutative and noncommutative rings
Chris Conidis, University of Waterloo, Canada


We 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.

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Capturing degrees of definability via models of set theory
Grigor Sargsyan, University of California at Los Angeles, USA


We 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.

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Partial functions and domination
Frank Stephan, National University of Singapore


The 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.

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Coherent extender sequence
Hugh Woodin, University†of†California at Berkeley, USA
National University of Singapore



We present the Martin-Steel theory of extender models and prove comparison assuming nice-UBH.

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Where comparison fails
Hugh Woodin, University†of†California at Berkeley, USA
National University of Singapore



We show that the Martin-Steel theory cannot be extended past the level of superstrong cardinals.

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Supercompact cardinals
Hugh Woodin, University†of†California at Berkeley, USA
National 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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