Diophantine Sets, Primes, and the Resolution of Hilbert’s 10th Problem

The term recursively enumerable set has its roots in logic but today is most commonly seen in reference to the theory of Turing machines. The Turing machine model is conceptually a very simple one for an abstract computing device but has been proved to be so powerful that many believe that that which is “computable” or “recursive” in any reasonable sense of the word is computable by some Turing machine; this is commonly known as Church’s Thesis. For the purposes of our discussion, we need only sketch the model: A Turing machine consists of an infinite tape with discrete “squares,” and a read/write head that can move left and right along the tape as well as read and write symbols in the squares on the tape. The rules that the head is allowed to use can only direct it to either write a new symbol on the tape or move precisely one square to the left or right. A Turing machine may reach a “halting state” – in this case, the machine simply stops – or it may operate indefinitely without ever reaching such a state. It is this last condition that gives rise to the notion of a recursively enumerable set. We can set a canonical method for representing a tuple (x1, . . . , xn) using symbols on a Turing machine tape, then “program” the head with rules that allow it to calculate based on this input. Then we say that a set E is recursively enumerable if there exists some Turing machine that, given (x1, . . . , xn) as input, will eventually halt if and only if (x1, . . . , xn)∈E. (The name “recursively enumerable” comes from the fact that any such set can actually be “enumerated” by a Turing machine by successively printing out all of its members onto the tape.) The Turing machine model was proposed and developed in the 1930’s by Alan Turing, Alonzo Church, and Kurt Gödel; they formulated the first rigorous definitions of the formerly intuitive notions of algorithm and computability. In doing so, they constructed the foundations upon which #10 of Hilbert’s 23 outstanding mathematical problems would eventually be solved. Thirty years before the concept of the Turing machine had been published, Hilbert had phrased the problem as follows: