N. C. A. DA COSTA AND F. A. DORIA SUPPES PREDICATES AND THECONSTRUCTION OF UNSOLVABLE PROBLEMS INTHE AXIOMATIZED SCIENCES ABSTRACT. We firstreviewour previous workonSuppespredicates andthe axiomati- zationof the empiricalsciences. We then state someundecidability andincompleteness results in classical analysis that lead to the explicit construction of expressions for characteristic functions in all complete arithmetical degrees. Out of those results we show that for any degree there are corresponding 'meaningful' unsolvable problems in any axiomatizedtheory that includes the language of classical analysis. Moreover wealso show that withinour formalizationthere are 'natural' unsolvableproblems and undecidablesentences whichare harder than any arithmeticproblem. As applications wediscuss a 1974 HilbertSymposiumproblemby Arnold on the existence of algo- rithms for the decision of properties of polynomial dynamical systems over Z, prove the incompleteness of the theory of finishNash games, and delve onrelatedquestions. Neither forcing nor diagonalizations are used in those constructions. 1. INTRODUCTION Suppes predicates were the starting point in the recent development of a technique for the construction of both algorithmically undecid- able sets of objects in physics and undecidable 'meaningful' sentences about physical objects. (See for details (da Costa, 1988; Suppes,1967, 1988).) That technique allowed us to prove the undecidability and incompleteness of most of classicaland quantum physics, providedthat they are givena first-order axiomatization (through Suppes predicates) that includes the language of classical elementary analysis (da Costa, 1991 a, 1991b, to appear, 1994 a; Stewart, 1991). Actual examples dealt with the proof of the incompleteness of chaos theory (da Costa, 1991 a) and with the related question of the existence of problems in dynam- ical systems theory whose solution is equivalent to solving very hard Diophantine problems, such as Fermat's Conjecture (da Costa, 1994 a). The same techniquereached beyondphysics and led to the proof of the incompleteness of the theory of Hamiltonian models for the dynamics of economical systems (Lewis, 1991b), while providing a partial result related to the recent proof by Lewis of the noncomputability of Arrow- Debreu equilibria (da Costa, 1992 d; Lewis, 1991 a). We can still list 151 P. Humphreys (cd.), Patrick Suppes: Scientific Philosopher, Vol. 2, 151-193. @ 1994 Kluwer Academic Publishers. Printedin theNetherlands.