Random reals from usual mathematical notions

BECHER, VERÓNICA

Nanjing

Conferencia; Computability, Complexity and Randomness; 2008

Institute of Mathematical Science, Nanjing University, China

We develop a general framework to provide examples of random reals arising from subsets of natural spaces in mathematicswhich can be represented as limit spaces of increasing sequences.Our examples are drawn from elementary analysis,and are defined from a standpoint of Computer Science:(1) we require a monotone Turing machine universal by adjunction--it can be argued that this is the most natural condition on universality--;(2) mathematical objects ought to be represented as limit elements of sequences output by a Turing machine performing infinite computations.From a given  partially ordered set D endowed with a computable partial order, we consider the limit space Dhat=Dup/ sim  of equivalence classes of increasing sequences over D, relative to the equivalence relation associated to  asymptotic equality.We consider several example spaces  Dhat.In particular, the space of the real numbers  -- augmented with -infty and +infty -- can be obtained as such a   Dhat  via the two classical ways:  as the limit space of its Dedekind cuts, and as the Cauchy completion of Q. More generally, the Cauchy completion of any metric space (X,d)-- augmented with open balls centered in X with rational radius --can be obtained as such a Dhat.