my forty years on his shoulders - Department of Mathematics

51Also consider the recursive unsolvability phenomena. Perhaps the moststriking example of this for the working mathematician is the recursiveunsolvability of Diophantine problems over the integers (Hilbert’s tenthproblem), as discussed in section 3. We have, at present, no idea of the boundarybetween recursive decidability and recursive undecidability in this realm. Yet Iconjecture that we will understand this in the future, and that we will find,perhaps, that recursive undecidability kicks in already for degree 4 with 4variables. However, this would require a complete overhaul of the currentsolution to Hilbert’s tenth problem, replete with new deep ideas. This wouldresult in a sharp increase in the level of interest for the working mathematicianwho is not particularly concerned with issues in the foundations of mathematics.In addition, we still do not know if there is an algorithm to decidewhether a Diophantine problem has a solution over the rationals. I conjecturethat this will be answered in the negative, and that the solution will involve someclever number theoretic constructions of independent interest for number theory.We now come to the future of the Incompleteness Phenomena. We haveseen how far this has developed thus far:i. First Incompleteness. Some incompleteness in the presence of somearithmetic. (Gödel 1931).ii. Second Incompleteness. Incompleteness concerning the most basicmetamathematical property - consistency. (Gödel 31), (Hilbert Bernays1934,1939), (Feferman 1960), (Boolos 93).iii. Consistency of the AxC. Consistency of the most basic, and oncecontroversial, early candidate for a new axiom of set theory. (Gödel 1940).