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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
Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 31
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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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