Ordinal numbers and the Hilbert basis theorem

Abstract

In [5] and [21] we studied countable algebra in the context of “reverse mathematics”. We considered set existence axioms formulated in the language of second order arithmetic. We showed that many well-known theorems about countable fields, countable rings, countable abelian groups, etc. are equivalent to the respective set existence axioms which are needed to prove them.One classical algebraic theorem which we did not consider in [5] and [21] is the Hilbert basis theorem. Let K be a field. For any natural number m, let K[x1,…,xm] be the ring of polynomials over K in m commuting indeterminates x1,…,xm. The Hilbert basis theorem asserts that for all K and m, every ideal in the ring K[x1,…,xm] is finitely generated. This theorem is of fundamental importance for invariant theory and for algebraic geometry. There is also a generalization, the Robson basis theorem [11], which makes a similar but more restrictive assertion about the ring K〈x1,…,xm〉 of polynomials over K in mnoncommuting indeterminates.In this paper we study a certain formal version of the Hilbert basis theorem within the language of second order arithmetic. Our main result is that, for any or all countable fields K, our version of the Hilbert basis theorem is equivalent to the assertion that the ordinal number ωω is well ordered. (The equivalence is provable in the weak base theory RCA0.) Thus the ordinal number ωω is a measure of the “intrinsic logical strength” of the Hilbert basis theorem. Such a measure is of interest in reference to the historic controversy surrounding the Hilbert basis theorem's apparent lack of constructive or computational content.