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REVERSE MATHEMATICS, YOUNG DIAGRAMS, AND THE ASCENDING CHAIN CONDITION

  • KOSTAS HATZIKIRIAKOU (a1) and STEPHEN G. SIMPSON (a2)

Abstract

Let S be the group of finitely supported permutations of a countably infinite set. Let $K[S]$ be the group algebra of S over a field K of characteristic 0. According to a theorem of Formanek and Lawrence, $K[S]$ satisfies the ascending chain condition for two-sided ideals. We study the reverse mathematics of this theorem, proving its equivalence over $RC{A_0}$ (or even over $RCA_0^{\rm{*}}$ ) to the statement that ${\omega ^\omega }$ is well ordered. Our equivalence proof proceeds via the statement that the Young diagrams form a well partial ordering.

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[1] Andrews, G. E., The Theory of Partitions, Cambridge University Press, New York, 2014.
[2] Aschenbrenner, M. and Yan Pong, W., Orderings of monomial ideals . Fundamenta Mathematicae, vol. 181 (2004), no. 1, pp. 2774.
[3] Chong, C.-T., Feng, Q., Slaman, T. A., and Woodin, W. H., editors, Infinity and Truth , Number 25 in IMS Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, World Scientific Publishing Co. Pte. Ltd., Singapore, 2014.
[4] Curtis, C. W. and Reiner, I., Representation Theory of Finite Groups and Associative Algebras, Interscience, Wiley, Hoboken, NJ, 1962.
[5] de Jongh, D. H. J. and Parikh, R., Well-partial orderings and hierarchies . Indagationes Mathematicae, vol. 80 (1977), no. 3, pp. 195207.
[6] Formanek, E. and Lawrence, J., The group algebra of the infinite symmetric group . Israel Journal of Mathematics, vol. 23 (1976), nos. 3 and 4, pp. 325331.
[7] Fulton, W., Young Tableaux With Applications to Representation Theory and Geometry, Number 35 in London Mathematical Society Student Texts, Cambridge University Press, New York, 1997.
[8] Hatzikiriakou, K., Algebraic disguises of ${\rm{\Sigma }}_1^0$ induction . Archive for Mathematical Logic, vol. 29 (1989), no. 1, pp. 4751.
[9] Hatzikiriakou, K., Commutative algebra in subsystems of second order arithmetic, Ph.D. thesis, Pennsylvania State University, 1989.
[10] Hatzikiriakou, K., A note on ordinal numbers and rings of formal power series . Archive for Mathematical Logic, vol. 33 (1994), no. 4, pp. 261263.
[11] Hilbert, D., Ueber die Theorie der algebraischen Formen . Mathematische Annalen, vol. 36 (1990), no. 4, pp. 473534.
[12] Hilbert, D., Über das Unendliche . Mathematische Annalen, vol. 95 (1926), no. 1, pp. 161190.
[13] Kerber, A., Representations of Permutation Groups I, Number 240 in Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1971.
[14] Kołodziejczyk, L. A. and Yokoyama, K., Categorical characterizations of the natural numbers require primitive recursion . Annals of Pure and Applied Logic, vol. 166 (2015), no. 2, pp. 219231.
[15] Maclagan, D., Antichains of monomial ideals are finite . Proceedings of the American Mathematical Society, vol. 129 (2000), no. 6, pp. 16091615.
[16] Mansfield, R. and Weitkamp, G., Recursive Aspects of Descriptive Set Theory, Oxford Logic Guides, Oxford University Press, Oxford, 1985.
[17] Marcone, A., WQO and BQO theory in subsystems of second order arithmetic. In [27], pp. 303–330, 2005.
[18] Nash-Williams, C. St. J. A.. On better-quasi-ordering transfinite sequences . Mathematical Proceedings of the Cambridge Philosophical Society, vol. 64 (1968), no. 2, pp. 273290.
[19] Noether, M., Gordan, Paul. Mathematische Annalen, vol. 75 (1914), no. 1, pp. 141.
[20] Robson, J. C., Well quasi-ordered sets and ideals in free semigroups and algebras . Journal of Algebra, vol. 55 (1978), no. 2, pp. 521535.
[21] Schmidt, D., Well-Partial Orderings and Their Maximal Order Types, Habilitationsschrift, Heidelberg University, Heidelberg, 1979.
[22] Simpson, S. G., BQO theory and Fraïssé’s conjecture. In [16], pp. 124–138, 1985.
[23] Simpson, S. G., Nichtbeweisbarkeit von gewissen kombinatorischen Eigenschaften endlicher Bäume . Archiv für mathematische Logik und Grundlagenforschung, vol. 25 (1985), no. 1, pp. 4565.
[24] Simpson, S. G., editor, Logic and Combinatorics, Contemporary Mathematics, American Mathematical Society, Providence, RI, 1987.
[25] Simpson, S. G., Ordinal numbers and the Hilbert basis theorem, this Journal, vol. 53 (1988), no. 3, pp. 961–974.
[26] Simpson, S. G., Partial realizations of Hilbert’s program, this Journal, vol. 53 (1988), no. 2, pp. 349–363.
[27] Simpson, S. G., editor, Reverse Mathematics 2001, Number 21 in Lecture Notes in Logic, Association for Symbolic Logic, 2005.
[28] Simpson, S. G., Subsystems of Second Order Arithmetic, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1999. Second Edition, Perspectives in Logic, Association for Symbolic Logic, Cambridge University Press, New York, 2009.
[29] Simpson, S. G., Baire categoricity and ${\rm{\Sigma }}_1^0$ induction . Notre Dame Journal of Formal Logic, vol. 55 (2014), no. 1, pp. 7578.
[30] Simpson, S. G., An objective justification for actual infinity? In [3], pp. 225–228, 2014.
[31] Simpson, S. G., Toward objectivity in mathematics. In [3], pp. 157–169, 2014.
[32] Simpson, S. G., Comparing WO $\left( {{\omega ^\omega }} \right)$ with ${\rm{\Sigma }}_2^0$ induction. arXiv:1508.02655, 11 August 2015. 6 pp.
[33] Simpson, S. G. and Smith, R. L., Factorization of polynomials and ${\rm{\Sigma }}_1^0$ induction . Annals of Pure and Applied Logic, vol. 31 (1986), pp. 289306.
[34] Simpson, S. G. and Yokoyama, K., Reverse mathematics and Peano categoricity . Annals of Pure and Applied Logic, vol. 164 (2013), no. 3, pp. 284293.
[35] Tait, W. W., Finitism . Journal of Philosophy, vol. 78 (1981), no. 9, pp. 524546.
[36] van Engelen, F, Miller, A. W., and Steel, J., Rigid Borel sets and better quasi-order theory . In [24], pp. 199222, 1987.
[37] Yokoyama, K., On the strength of Ramsey’s Theorem without ${{\rm{\Sigma }}_1}$ -induction . Mathematical Logic Quarterly, vol. 59 (2013), nos. 1 and 2, pp. 108111.
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