A Constructive Version of the Hilbert Basis Theorem

The Hilbert Basis Theorem was the first major example of a non-constructive proof recognized in mathematics. Gordan said, on the subject of the theorem, “das ist keine Mathematik, das ist Theologie!” — “this is not Mathematics, this is Theology!” [8] Although there are several equivalent statements of the theorem, in this paper we will consider the version which states, in essence, that all rings of polynomials over countable fields are finitely generated. (More generally, the theorem holds for polynomial ideals over any Notherian ring. All the proofs in this paper can easily be adapted to this more general situation.) In this paper, we will consider two different constructive proofs. Each is accomplished by applying Gödel’s Dialectica Interpretation to a classical proof of the theorem. Both yield algorithms that are instances of primitive recursive functionals of finite types, essentially a simple programming language in which one can only express total functions. The first, from a standard proof, yields a constructive version requiring higher-type primitive recursion. The second, obtained by applying the interpretation to a proof by Simpson [9], yields a more efficient algorithm in a sense which will be explained later. An overview of this thesis is as follows: In sections 2 and 3 we present logical and algebraic preliminaries, respectively. In section 4 we present our first proof of Dickson’s Lemma. In section 5 we show how to derive the Hilbert Basis Theorem from Dickson’s Lemma. Finally, in section 6 we give our second, more elegant constructive proof of Dickson’s Lemma.