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Wadge hierarchy and Veblen hierarchy Part I: Borel sets of finite rank

Published online by Cambridge University Press:  12 March 2014

J. Duparc
J. Duparc*
Affiliation:
Equipe de Logique Mathématique, CNRS URA 753, et Université Paris VII, U.F.R. de Mathématiques, 2 Place Jussieu. 75251 Paris Cedex 05, France, E-mail: [email protected]
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Abstract

We consider Borel sets of finite rank A ⊆ ∧ω where cardinality of Λ is less than some uncountable regular cardinal . We obtain a “normal form” of A, by finding a Borel set Ω, such that A and Ω continuously reduce to each other. In more technical terms: we define simple Borel operations which are homomorphic to ordinal sum, to multiplication by a countable ordinal, and to ordinal exponentiation of base , under the map which sends every Borel set A of finite rank to its Wadge degree.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 1 , March 2001 , pp. 56 - 86
Copyright
Copyright © Association for Symbolic Logic 2001

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References

REFERENCES

[1]Kechris, A.S., Classical descriptive set theory, Graduate texts in mathematics, vol. 156, Springer Verlag, 1994.Google Scholar
[2]Louveau, A., Some results in the Wadge hierarchy of Borel sets, Cabal seminar 79-81, Lecture Notes in Mathematics, vol. 1019, Springer Verlag, 1983, pp. 28–55.CrossRefGoogle Scholar
[3]Louveau, A. and Raymond, J. St., The strength of Borel Wadge determinacy, Cabal seminar 81–85, Lecture Notes in Mathematics, vol. 1333, Springer Verlag, 1985, pp. 1–30.Google Scholar
[4]Louveau, A. and Raymond, J. St., Les propriétés de réduction et de norme pour les classes de Boréliens, Fundamenta Mathematical vol. 131 (1988), pp. 223–243.CrossRefGoogle Scholar
[5]Martin, D. A., Borel determinacy, Annals of Mathematics, vol. 102 (1975), pp. 363–371.CrossRefGoogle Scholar
[6]van Wesep, R., Wadge degrees and descriptive set theory, Cabal seminar 76–77, (Proceedings of the Caltech-UCLA logic seminar, 1976–77), Lecture Notes in Mathematics, vol. 689, Springer, Berlin, 1978, pp. 151–170.Google Scholar
[7]Veblen, O., Continuous increasing fonctions of finite and transfinite ordinals, Transactions of the American Mathematical Society, vol. 9 (1908), pp. 280–292.CrossRefGoogle Scholar
[8]Wadge, W. W., Degrees of complexity of subsets of the Baire space, Notices of the American Mathematical Society, (1972), pp. A–714.Google Scholar
[9]Wadge, W. W., Reducibility and determinateness on the Baire space., Ph.D. thesis, Berkeley, 1984.Google Scholar