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Ordinal numbers and the Hilbert basis theorem

Published online by Cambridge University Press:  12 March 2014

Stephen G. Simpson
Stephen G. Simpson*
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
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Extract

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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 53 , Issue 3 , September 1988 , pp. 961 - 974
Copyright
Copyright © Association for Symbolic Logic 1988

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References

REFERENCES

[1]Bell, E. T., The development of mathematics, McGraw-Hill, New York, 1940.Google Scholar
[2]Buchholz, W. and Wainer, S., Provably computable functions and the fast growing hierarchy, in [16], pp. 179–198.Google Scholar
[3]De Jongh, D. H. J. and Parikh, R., Weil partial orderings and hierarchies, Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings, Series A, vol. 80 = Indagationes Mathematicae, vol. 39 (1977), pp. 195–207.Google Scholar
[4]van Engelen, F., Miller, A. W. and Steel, J., Rigid Borel sets and better quasiorder theory, in [16], pp. 199–222.Google Scholar
[5]Friedman, H., Simpson, S. G. and Smith, R. L., Countable algebra and set existence axioms, Annals of Pure and Applied Logic, vol. 25 (1983), pp. 141–181; addendum, vol. 28 (1985), pp. 320–321.CrossRefGoogle Scholar
[6]Gordan, P., Neuer Beweis des Hilbert'schen Satzes über homogene Funktionen, Nachrichten von der Königliche Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 1899, pp. 240–242.Google Scholar
[7]Gordan, P., Les invariantes des formes binaires, Journal de Mathématiques Pures et Appliquées, ser. 5, vol. 6 (1900), pp. 141–156.Google Scholar
[8]Noether, M., Gordan, Paul, Mathematische Annalen, vol. 75 (1914), pp. 1–45.CrossRefGoogle Scholar
[9]Parsons, C., On a number-theoretic choice schema and its relation to induction, Intuitionism and proof theory (Myhill, J. et al, editors), North-Holland, Amsterdam, 1970, pp. 459–473.Google Scholar
[10]Robson, J. C., Polynomials satisfied by matrices, Journal of Algebra, vol. 55 (1978), pp. 509–520.CrossRefGoogle Scholar
[11]Robson, J. C., Well quasi-ordered sets and ideals in free semigroups and algebras, Journal of Algebra, vol. 55 (1978), pp. 521–535.CrossRefGoogle Scholar
[12]Schmidt, D., Well-partial-orderings and their maximal order types, Habilitationsschrift, Heidelberg, 1979.Google Scholar
[13]Schütte, K. and Simpson, S. G., Ein in der reinen Zahlentheorie unbeweisbarer Satz über endliche Folgen von natürlichen Zahlen, Archiv für Mathematische Logik und Grundlagenforschung, vol. 25 (1985), pp. 75–89.CrossRefGoogle Scholar
[14]Sieg, W., Fragments of arithmetic, Annals of Pure and Applied Logic, vol. 28 (1985), pp. 33–71.CrossRefGoogle Scholar
[15]Simpson, S. G., BQO theory and Fraïsse's conjecture, Chapter 9 in Mansfield, R. and Weitkamp, G., Recursive aspects of descriptive set theory, Oxford University Press, Oxford, 1985, pp. 124–138.Google Scholar
[16]Simpson, S. G. (editor), Logic and combinatorics, Contemporary Mathematics, vol. 65, American Mathematical Society, Providence, Rhode Island, 1987.CrossRefGoogle Scholar
[17]Simpson, S. G., Subsystems of second order arithmetic, in preparation.Google Scholar
[18]Simpson, S. G., Subsystems of Z2 and reverse mathematics, appendix to G. Takeuti, Proof theory, 2nd ed, North-Holland, Amsterdam, 1986, pp. 434–448.Google Scholar
[19]Simpson, S. G., Unprovable theorems and fast-growing functions, in [16], pp. 359–394.CrossRefGoogle Scholar
[20]Simpson, S. G., Which set existence axioms are needed to prove the Cauchy/Peano theorem of ordinary differential equations? this Journal, vol. 49 (1984), pp. 783–802.Google Scholar
[21]Simpson, S. G. and Smith, R. L., Factorization of polynomials and 1Σ induction, Annals of Pure and Applied Logic, vol. 31 (1986), pp. 289–306.CrossRefGoogle Scholar
[22]Statman, R., Well partial orderings, ordinals and trees, preprint, Rutgers University, 11, 1980, 13 pages.Google Scholar