Seminários de Lógica Matemática

8 de Maio 2008, 17h00

A NOTE ON THE REPRESENTATION OF THE REALS
SALA 3.10 - IST

Patrícia Engrácia (U LISBOA)

Abstract:

There are many classical constructions of real numbers, such as Cauchy sequences of rational numbers, Dedekind cuts in the field of rationals, binary representations, signed-digit representations, and so on. Classically all these representations are equivalent, in the sense that they give rise to isomorphic structures.
In 1937, Turing introduced the first notion of computable real number. We would like to answer the following question: can we have an effective construction of the real numbers which preserves the notion of majorizability? We that there is a function g: N -> N such that, if f: N -> N represents a real number in the interval [-n,n], then f is bounded by g(n). As far as we know, none of the constructions above which are computable satisfy this majorizability condition.
In this talk, we present the modified signed-digit representation. This is a modification of the signed-digit representation which satisfies the desired property. We also show that this new representation is effectively equivalent to the Cauchy sequence representation.

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