my forty years on his shoulders - Department of Mathematics

12THEOREM 3.1. Let T be a consistent extension of Q in a relational type in manysorted predicate calculus of arbitrary cardinality. The sets of all existentialsentences in L(Q), with bounded universal quantifiers allowed, that are i)provable in T, ii) refutable in T, iii) provable or refutable in T, are each notrecursive.For the proof, see (Robinson 1952), and (Tarski, Mostowski, Robinson1953). It uses the construction of recursively inseparable recursively enumerablesets; e.g., {n: ϕ n (n) = 0} and {n: ϕ n (n) = 1}.One can obtain the following strong form of first incompleteness as animmediate Corollary.THEOREM 3.2. Let T be a consistent extension of Q in many sorted predicatecalculus whose relational type and axioms are recursively enumerable. There isan existential sentence in L(Q), with bounded universal quantifiers allowed, thatis neither provable nor refutable in T.We can use the negative solution to Hilbert’s tenth problem in order toobtain other forms of first incompleteness that are stronger in certain respects. Infact, Hilbert’s tenth problem is still a great source of very difficult problems onthe border between logic and number theory, which we will discuss below.Hilbert asked for a decision procedure for determining whether a givenpolynomial with integer coefficients in several integer variables has a zero.The problem received a negative answer in 1970 by Y. Matiyasevich,building heavily on earlier work of J. Robinson, M. Davis, and H. Putnam. It iscommonly referred to as the MRDP theorem (in reverse historical order). See