quinta-feira, janeiro 08, 2009

Números irreais

Acabei de ler o paper do Chaitin:

How real are real numbers?

Authors: G. J. Chaitin (IBM Research)
Abstract: We discuss mathematical and physical arguments against continuity and in favor of discreteness, with particular emphasis on the ideas of Emile Borel (1871-1956).

Nao conhecia esse paradoxo, descrito no paper. É divertido:

Richard’s paradox: Diagonalize over all nameable reals −! a nameable, unnameable real

The problem is that the set of reals is uncountable, but the set of all possible
texts in English or French is countable, and so is the set of all possible
mathematical definitions or the set of all possible mathematical questions,
since these also have to be formulated within a language, yielding at most
a denumerable infinity of possibilities. So there are too many reals, and not
enough texts.

The first person to notice this difficulty was Jules Richard in 1905, and the
manner in which he formulated the problem is now called Richard’s paradox.
Here is how it goes. Since all possible texts in French (Richard was
French) can be listed or enumerated, a first text, a second one, etc.,2 you
can diagonalize over all the reals that can be defined or named in French and
produce a real number that cannot be defined and is therefore unnameable.
However, we’ve just indicated how to define it or name it!

In other words, Richard’s paradoxical real differs from every real that
is definable in French, but nevertheless can itself be defined in French by
specifying in detail how to apply Cantor’s diagonal method to the list of all
possible mathematical definitions for individual real numbers in French!

How very embarrassing! Here is a real number that is simultaneously
nameable yet at the same time it cannot be named using any text in French.

Sobre o Teorema do Macaco Infinito, ver aqui e aqui.

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