Indecidability and Incompleteness In Formal Axiomatics as Questioned by Anticipatory Processes

Hilbert's conjecture that the whole of mathematics could be provided by a finite set of axioms (Hilbert, publ. 1980) was challenged in branches of mathematics, devoted to arithmetics and algorithmic computation, by Gödel (1931), Church (1936), Turing (1937), and Chaitin (1998). This questioned what can be expected from scientific knowledge, in particular through the mesh of mathematical certainty, in the assessment of what could be considered true about our universe, that is also on ourselves via self-evaluation possibility.

This study will thus revisit some current problems about the conditions required for allowing a measure of "something" likeky unknown, situated "somewhere", in terms of distances and dimensions. The debate will then focus on the scope of mathematical knowledge, with special regards to indecidability, incompleteness, and the fate of such mathematical realites claimed to escape the field of mathematics, like for Chaitin's 'omega number'. The formal involvement of anticipatory processes in finding solutons through biological self-evaluaton will be analyzed in several steps.