Hilbert’s 10th problem and incompleteness
The negative solution of Hilbert’s 10th problem is one of the most striking manifestations of the incompleteness of arithmetic. Undecidable propositions don’t have to look “paradoxical” like the Gödel statement. And they don’t have to involve unimaginably large structures like the Goodstein sequence.
There are undecidable statements that simply talk about existence of solutions of Diophantine equations.
The reason for that is even more striking: any question of the form “is there a natural number possessing the property P” can be formulated as “does this particular Diophantine equation have a solution”. Basically, any property of the natural numbers can be expressed as a polynomial equation. And we are not talking about some unimaginably huge polynomials here. First of all, they are effectively defined, i.e. we can really construct them. And also their size is bounded by some very reasonable constants. E.g. we know that we can use equations of the degree not greater than 4 with no more than 58 unknowns. Or we can limit the number of unknowns to 26 so we can use the letters of the alphabet, in which case we would need to allow the degree to go up to 24. Even 9 unknowns are sufficient, but then the degree would have to grow up to 1045 .
I can’t help noticing once again that arithmetic has way way more expressive power than we actually need.
The history of Hilbert’s 10th is as fascinating as its contents.
It is curious that Julia Robinson’s teacher was Alfred Tarski who strongly believed that there exist non-Diophantine enumerable sets. And Martin Davis who formulated the key conjecture was a student of Alonzo Church. Yuri Matiyasevich (who gave the final proof of that conjecture now known as the DPRM theorem) in one of his lectures mentioned that he had used the Russian word наглый to describe Davis’s conjecture. This word is actually much stronger than “bold”, it’s almost “arrogant-obnoxious-badass-bold”.
I find it remarkable that this astonishing solution of a genuine number-theoretical problem was achieved by logicians. BTW, the “P” in “DPRM” is Hilary Putnam who is more of a philosopher than a logician.
Wikipedia does a good job too. Here and here. One small addition. Some texts give the impression that the DPRM theorem solves Hilbert’s 10th “because undecidable sets are known to exist”. In fact we don’t need to know about undecidable sets or to invoke the halting argument. We can simply use the same Turing’s diagonalization trick as for the halting problem. Since we can enumerate all diophantine equations, we can diagonalize by feeding the n-th equation with n as the parameter to the “oracle”. Then, since the output is a diophantine set, its representation must be on the original list, and we can show that the “oracle” must lie…