Fri Feb 1, 2008
60 Evans Hall, 4:10–6 PM
Theodore Slaman (Mathematics, UC Berkeley)
Effective Randomness and Continuous Measures
We will elaborate on the question, “For which reals x does there exist a measure m such that x is effectively random relative to m?” which Jan Reimann discussed in the Logic Colloquium last semester. We will review what is known about the general question. We give several conditions on x equivalent to there being a continuous measure which makes x random. We show that for all but countably many reals x these conditions apply, so there is a continuous measure which makes x random. There is a metamathematical aspect of this investigation. As one requires higher arithmetic levels in the degree of randomness, one must make use of more iterates of the power set of the continuum. We will then focus on questions of 1-randomness and those x ‘s which are recursive in 0’.