Journal of Symbolic Logic

The Steel hierarchy of ordinal valued Borel mappings

J. Duparc

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Abstract

Given well ordered countable sets of the form $\lamphi$, we consider Borel mappings from $\lamphiom$ with countable image inside the ordinals. The ordinals and $\lamphiom$ are respectively equipped with the discrete topology and the product of the discrete topology on $\lamphi$. The Steel well-ordering on such mappings is defined by $\phi\minf\psi$ iff there exists a continuous function $f$ such that $\phi(x)\leq\psi\circ f(x)$ holds for any $x\in\lamphiom$. It induces a hierarchy of mappings which we give a complete description of. We provide, for each ordinal $\alpha$, a mapping $\T{\alpha}$ whose rank is precisely $\alpha$ in this hierarchy and we also compute the height of the hierarchy restricted to mappings with image bounded by $\alpha$. These mappings being viewed as partitions of the reals, there is, in most cases, a unique distinguished element of the partition. We analyze the relation between its topological complexity and the rank of the mapping in this hierarchy.

Article information

Source
J. Symbolic Logic, Volume 68, Issue 1 (2003), 187-234.

Dates
First available in Project Euclid: 21 February 2003

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1045861511

Digital Object Identifier
doi:10.2178/jsl/1045861511

Mathematical Reviews number (MathSciNet)
MR1959317

Zentralblatt MATH identifier
1050.03032

Citation

Duparc, J. The Steel hierarchy of ordinal valued Borel mappings. J. Symbolic Logic 68 (2003), no. 1, 187--234. doi:10.2178/jsl/1045861511. https://projecteuclid.org/euclid.jsl/1045861511


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References

  • C. Chalons A unique integer associated to each map from $E^\omega$ to $\omega$, Comptes Rendus de l' Académie des Sciences. Série I Mathématique, vol. 331 (2000), no. 7, pp. 501--506.
  • -------- Veblen Hierarchy and Wadge Hierarchy. Part II: Borel sets of infinite rank, Journal of Symbolic Logic, submitted.
  • F. van Engelen, A. Miller, and J. Steel Rigid Borel sets and better quasi-order theory, Logic and combinatorics (Arcata, Calif., 1985), American Mathematical Society, Providence, RI,1987, pp. 199--222.
  • D. Gale and F.M. Stewart Infinite games with perfect information, Annals of Mathematics Studies, vol. 28 (1953), pp. 245--266.
  • A. Louveau Some results in the Wadge hierarchy of Borel sets, Cabal seminar 79--81, Springer, Berlin,1983, Lecture Notes in Mathematics (1019), pp. 28--55.
  • A. Louveau and J. Saint-Raymond Les propriétés de réduction et de norme pour les classes de Boréliens, Fundamenta Mathematicae, vol. 131 (1988), pp. 223--243.
  • D.A. Martin Borel Determinacy, Annals of Mathematics, vol. 102 (1975), pp. 363--371.
  • M. Sendak Where the wild things are, Harper & Row, New York,1963.
  • W.W. Wadge Degrees of complexity of subsets of the Baire space, Notice of the American Mathematical Society,1972, A-714.