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A CLASSIFICATION OF THE WADGE HIERARCHIES ON ZERO-DIMENSIONAL POLISH SPACES

Part of: Set theory Spaces with richer structures

Published online by Cambridge University Press:  19 May 2023

RAPHAËL CARROY Open the ORCID record for RAPHAËL CARROY [Opens in a new window] ,
LUCA MOTTO ROS Open the ORCID record for LUCA MOTTO ROS [Opens in a new window]  and
SALVATORE SCAMPERTI
RAPHAËL CARROY*
Affiliation:
DIPARTIMENTO DI MATEMATICA «GIUSEPPE PEANO» UNIVERSITÀ DI TORINO VIA CARLO ALBERTO 10, 10123 TORINO, ITALY E-mail: luca.mottoros@unito.it E-mail: salvatore.scamperti@unito.it
LUCA MOTTO ROS
Affiliation:
DIPARTIMENTO DI MATEMATICA «GIUSEPPE PEANO» UNIVERSITÀ DI TORINO VIA CARLO ALBERTO 10, 10123 TORINO, ITALY E-mail: luca.mottoros@unito.it E-mail: salvatore.scamperti@unito.it
SALVATORE SCAMPERTI
Affiliation:
DIPARTIMENTO DI MATEMATICA «GIUSEPPE PEANO» UNIVERSITÀ DI TORINO VIA CARLO ALBERTO 10, 10123 TORINO, ITALY E-mail: luca.mottoros@unito.it E-mail: salvatore.scamperti@unito.it
*
E-mail: raphael.carroy@unito.it
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Abstract

We provide a complete classification, up to order-isomorphism, of all possible Wadge hierarchies on zero-dimensional Polish spaces using (essentially) countable ordinals as complete invariants. We also observe that although our assignment of invariants is very simple and there are only $ \aleph _1 $-many equivalence classes, the above classification problem is quite complex from the descriptive set-theoretic point of view: in particular, there is no Borel procedure to determine whether two zero-dimensional Polish spaces have isomorphic Wadge hierarchies. All results are based on a complete and explicit description of the Wadge hierarchy on an arbitrary zero-dimensional Polish space, depending on its topological properties.

Keywords

Wadge reducibilitycontinuous reducibilityWadge quasi-orderzero-dimensional Polish spaces

MSC classification

Primary: 03E15: Descriptive set theory
Secondary: 54E50: Complete metric spaces

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Type
Article
Information
The Journal of Symbolic Logic , Volume 90 , Issue 2 , June 2025 , pp. 887 - 917
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

Andretta, A., Equivalence between Wadge and Lipschitz determinacy . Annals of Pure and Applied Logic, vol. 123 (2003), nos. 1–3, pp. 163–192.CrossRefGoogle Scholar
Andretta, A., More on Wadge determinacy . Annals of Pure and Applied Logic, vol. 144 (2006), nos. 1–3, pp. 2–32.CrossRefGoogle Scholar
Andretta, A., The SLO principle and the Wadge hierarchy , Foundations of the Formal Sciences V (S. Bold, B. Löwe, and T. Räsch, editors), Studies in Logic (London), 11, College Publications, London, 2007, pp. 1–38.Google Scholar
Andretta, A. and Camerlo, R., The descriptive set theory of the Lebesgue density theorem . Advances in Mathematics, vol. 234 (2013), pp. 1–42.CrossRefGoogle Scholar
Andretta, A. and Martin, D. A., Borel–Wadge degrees . Fundamenta Mathematicae, vol. 177 (2003), no. 2, pp. 175–192.CrossRefGoogle Scholar
Camerlo, R., Continuous reducibility: Functions versus relations . Reports on Mathematical Logic, vol. 54 (2019), pp. 45–63.CrossRefGoogle Scholar
Camerlo, R., Marcone, A., and Motto Ros, L., On isometry and isometric embeddability between ultrametric Polish spaces . Advances in Mathematics, vol. 329 (2018), pp. 1231–1284.CrossRefGoogle Scholar
Camerlo, R. and Massaza, C., The Wadge hierarchy on Zariski topologies . Topology and its Applications, vol. 294 (2021), p. 107661.CrossRefGoogle Scholar
Carroy, R., A quasi-order on continuous functions, this Journal, vol. 78 (2013), no. 2, pp. 633–648.Google Scholar
Carroy, R., Medini, A., and Müller, S., Every zero-dimensional homogeneous space is strongly homogeneous under determinacy . Journal of Mathematical Logic, vol. 20 (2020), no. 3, p. 2050015.CrossRefGoogle Scholar
Carroy, R., Medini, A., and Müller, S., Constructing Wadge classes . Bulletin of Symbolic Logic, vol. 28 (2022), no. 2, pp. 207–257.CrossRefGoogle Scholar
Duparc, J., Wadge hierarchy and Veblen hierarchy. I. Borel sets of finite rank, this Journal, vol. 66 (2001), no. 1, pp. 56–86.Google Scholar
Duparc, J. and Vuilleumier, L., The Wadge order on the Scott domain is not a well-quasi-order, this Journal, vol. 85 (2020), no. 1, pp. 300–324.Google Scholar
Hertling, P., Topologische Komplexitätsgrade von Funktionen mit endlichem Bild , Ph.D. thesis, Fernuniversität Hagen, 1993.Google Scholar
Ikegami, D., Schlicht, P., and Tanaka, H., Borel subsets of the real line and continuous reducibility . Fundamenta Mathematicae, vol. 244 (2019), no. 3, pp. 209–241.CrossRefGoogle Scholar
Jech, T., Set Theory, Springer Monographs in Mathematics, Springer, Berlin, 2003.Google Scholar
Kechris, A. S., Classical Descriptive Set Theory, Graduate Texts in Mathematics, vol. 156, Springer, New York, 1995.CrossRefGoogle Scholar
Kihara, T. and Montalbán, A., On the structure of the Wadge degrees of BQO-valued Borel functions . Transactions of the American Mathematical Society, vol. 371 (2019), no. 11, pp. 7885–7923.CrossRefGoogle Scholar
Louveau, A., Some results in the Wadge hierarchy of Borel sets , Cabal Seminar 79–81 (Kechris, A. S., Martin, D. A., and Moschovakis, Y. N., editors), Lecture Notes in Mathematics, vol. 1019, Springer, Berlin, 1983, pp. 28–55.CrossRefGoogle Scholar
Louveau, A. and Saint-Raymond, J., The strength of Borel Wadge determinacy , Cabal Seminar 81–85 (Kechris, A. S., Martin, D. A., and Steel, J. R., editors), Lecture Notes in Mathematics, vol. 1333, Springer, Berlin, 1988, pp. 1–30.Google Scholar
Motto Ros, L., Borel-amenable reducibilities for sets of reals, this Journal, vol. 74 (2009), pp. 27–49.Google Scholar
Motto Ros, L., Baire reductions and good Borel reducibilities, this Journal, vol. 75 (2010), no. 1, pp. 323–345.Google Scholar
Motto Ros, L., Beyond Borel-amenability: Scales and superamenable reducibilities . Annals of Pure and Applied Logic, vol. 161 (2010), no. 7, pp. 829–836.CrossRefGoogle Scholar
Motto Ros, L., Schlicht, P., and Selivanov, V., Wadge-like reducibilities on arbitrary quasi-polish spaces . Mathematical Structures in Computer Science, vol. 25 (2015), no. 8, pp. 1705–1754.CrossRefGoogle Scholar
Pequignot, Y., A Wadge hierarchy for second countable spaces . Archive for Mathematical Logic, vol. 54 (2015), nos. 5–6, pp. 659–683.CrossRefGoogle Scholar
Schlicht, P., Continuous reducibility and dimension of metric spaces . Archive for Mathematical Logic, vol. 57 (2018), nos. 3–4, pp. 329–359.CrossRefGoogle Scholar
Selivanov, V. L., Boolean hierarchies of partitions over a reducible base . Algebra Logika, vol. 43 (2004), no. 1, pp. 77–109.CrossRefGoogle Scholar
Selivanov, V. L., Variations on the Wadge reducibility [translation of mat. Tr. 8 (2005), no. 1, 135–175; mr1955025] . Siberian Advances in Mathematics, vol. 15 (2005), no. 3, pp. 44–80.Google Scholar
Solovay, R. M., The independence of DC from AD , Cabal Seminar 76–77 (Kechris, A. S. and Moschovakis, Y. N., editors), Lecture Notes in Mathematics, vol. 689, Springer, Berlin, 1978, pp. 171–183.CrossRefGoogle Scholar
Steel, J. R., Determinateness and subsystems of analysis , Ph.D. thesis, University of California, Berkeley, 1977.Google Scholar
Van Wesep, R., Subsystems of second-order arithmetic, and descriptive set theory under the axiom of determinateness , Ph.D. thesis, University of California, Berkeley, 1977.Google Scholar
Van Wesep, R., Wadge degrees and descriptive set theory , Cabal Seminar 76–77 (Kechris, A. S. and Moschovakis, Y. N., editors), Lecture Notes in Mathematics, vol. 689, Springer, Berlin, 1978, pp. 151–170.CrossRefGoogle Scholar
Wadge, W. W., Reducibility and determinateness on the Baire space. Ph.D. thesis, University of California, Berkeley, 1983.Google Scholar