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A Constructive Version of the Hilbert Basis Theorem

Summary: A Constructive Version of the
Hilbert Basis Theorem
Aaron Hertz
May 7, 2004
1 Introduction
The Hilbert Basis Theorem was the first major example of a non-constructive
proof recognized in mathematics. Gordan said, on the subject of the theo-
rem, "das ist keine Mathematik, das ist Theologie!" -- "this is not Math-
ematics, this is Theology!" [8] Although there are several equivalent state-
ments 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 gen-
erated. (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 GĻodel's Dialectica Interpretation to a classical
proof of the theorem. Both yield algorithms that are instances of primi-
tive 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


Source: Avigad, Jeremy - Departments of Mathematical Sciences & Philosophy, Carnegie Mellon University


Collections: Multidisciplinary Databases and Resources; Mathematics