Classes of Polish Spaces Under Effective Borel Isomorphism

102 Pages·2016·0.697 MB·English

by  Vassilios Gregoriades

MEMOIRS
of the
American Mathematical Society
Volume 240 • Number 1135 (first of 5 numbers) • March 2016
Classes of Polish Spaces Under
Effective Borel Isomorphism
Vassilios Gregoriades
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
MEMOIRS
of the
American Mathematical Society
Volume 240 • Number 1135 (first of 5 numbers) • March 2016
Classes of Polish Spaces Under
Effective Borel Isomorphism
Vassilios Gregoriades
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
Providence, Rhode Island
Library of Congress Cataloging-in-Publication Data
Names: Gregoriades,Vassilios,1981–
Title: ClassesofPolishspacesundereffectiveBorelisomorphism/VassiliosGregoriades.
Description: Providence,RhodeIsland: AmericanMathematicalSociety,2016. —Series: Mem-
oirsoftheAmericanMathematicalSociety,ISSN0065-9266;volume240,number1135—Includes
bibliographicalreferencesandindex.
Identifiers: LCCN2015045901—ISBN9781470415631(alk. paper)
Subjects: LCSH:Polishspaces(Mathematics)—Metricspaces. —Isomorphisms(Mathematics)
Classification: LCC QA611.28 .G74 2016 — DDC 511.3/22–dc23 LC record available at
http://lccn.loc.gov/2015045901
DOI:http://dx.doi.org/10.1090/memo/1135
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Contents
Preface vii
Chapter 1. Introduction 1
1.1. A few words about the setting 1
1.2. The basic notions 3
1.3. Theorems in the general context 9
1.4. The Cantor-Bendixson decomposition 10
Chapter 2. The spaces NT 13
2.1. Definition and properties 13
2.2. Elementary facts about the classes of Δ1 isomorphism 19
1
Chapter 3. Kleene spaces 23
3.1. Definition and basic properties 23
3.2. Expanding the toolbox 26
3.3. Chains and antichains in Kleene spaces under (cid:3) 28
Δ1
1
3.4. Analogy with recursive pseudo-well-orderings 33
3.5. Incomparable hyperdegrees in Kleene spaces 38
Chapter 4. Characterizations of N up to Δ1 isomorphism 41
1
4.1. Copies of the complete binary tree 41
4.2. A characterization in terms of the perfect kernel 45
4.3. A measure-theoretic characterization 46
4.4. The tree of attempted embeddings 47
Chapter 5. Spector-Gandy spaces 51
5.1. The Spector-Gandy Theorem 51
5.2. Application to the spaces NT 57
5.3. Chains and antichains in Spector-Gandy spaces under (cid:3) 62
Δ1
1
Chapter 6. Questions and related results 71
6.1. Parametrization 71
6.2. Kleene spaces 71
6.3. Incomparable and minimal hyperdegrees 73
6.4. Connections with lattice theory 74
6.5. Attempted Embeddings and incomparability 75
6.6. Spector-Gandy spaces 77
6.7. Connections with Π1 equivalence relations 78
1
Bibliography 81
Index 85
iii

Abstract
We study the equivalence classes under Δ1 isomorphism, otherwise effective
1
Borel isomorphism, between complete separable metric spaces which admit a re-
cursive presentation and we show the existence of strictly increasing and strictly
decreasing sequences as well as of infinite antichains under the natural notion of
Δ1-reduction, as opposed to the non-effective case, where only two such classes
1
exist, the one of the Baire space and the one of the naturals. A key tool for our
study is a mapping T (cid:4)→ NT from the space of all trees on the naturals to the
class ofPolish spaces, for which everyrecursively presentedspace isΔ1-isomorphic
1
to some NT for a recursive T, so that the preceding spaces are representatives for
the classes of Δ1 isomorphism. We isolate two large categories of spaces of the
1
type NT, the Kleene spaces and the Spector-Gandy spaces and we study them
extensively. Moreover we give results about hyperdegrees in the latter spaces and
characterizations of the Baire space up to Δ1 isomorphism.
1
ReceivedbytheeditorJune23,2013and,inrevisedform,January20,2014.
ArticleelectronicallypublishedonOctober9,2015.
DOI:http://dx.doi.org/10.1090/memo/1135
2010 MathematicsSubjectClassification. Primary03E15,03D30,03D60.
Keywordsandphrases. Recursivelypresentedmetricspace,effectiveBorelisomorphism,Δ1
1
isomorphism,Δ1 injection,Kleenespace,Spector-Gandyspace.
1
(cid:2)c2015 American Mathematical Society
v

Preface
This article contains work carried between 2009 and 2013. It includes parts of
my Ph.D. Thesis, which was submitted to and approved by the Mathematics De-
partment of the University of Athens Greece in 2009 and supervised by Y. N.
Moschovakis and A. Tsarpalias. On a rough estimate these parts include
most of the material in the first two Chapters, the first Section in Kleene spaces
as well as results about the Cantor-Bendixson decomposition of Spector-Gandy
spaces−although the latter spaces are not identified with this name in my Thesis.
Theextendedabstractofaveryearlyversionofthisarticlehasbeenpublished
in the proceedings of the 8th Panhellenic Logic Symposium in 2011, Ioannina,
Greece.
I would like to express my gratitude to my supervisors and especially to Y. N.
Moschovakisforhisinvaluableguidancethroughthewholeprocessofpreparation
of this article.
TherealizationofthisprojectwouldnotbepossiblewithoutU. Kohlenbach,
who, aside from his valuable advice, has offered to the author a long term post-
doctoral position in the Logic Group at TU Darmstadt. I would like to take the
opportunity to express my sincere thanks to him.
The list of people who have helped me in this task contains one prominent
entry, my wife Stella, who knows first hand the difficulties of being the spouse of
a researcher in mathematics. I am deeply grateful to her for her enduring support
and encouragement.
Vassilis Gregoriades
March 2014
Darmstadt, Germany
vii

CHAPTER 1
Introduction
Thetopicofourresearchistheclassofcompleteseparablemetricspaceswhich
admitarecursivepresentationandtheirclassificationunderΔ1 (otherwiseeffective
1
Borel) isomorphism. Following [Mos09] we refer to the latter spaces simply as
recursively presented metric spaces. In the sequel we will explain how this can
be carried out in all Polish spaces. We begin this introductory chapter with a
discussion about the motivation and a short description of this article. Then we
proceed to the necessary definitions and basic theorems.
1.1. A few words about the setting
Thisarticleoriginatesfromthefollowingsimpleremarkaboutthedevelopment
of effective descriptive set theory as it is presented in the fundamental textbook
[Mos09], which is perhaps easy to overlook: one assumes that every recursively
presentedmetricspacewhichisnotdiscretehasnoisolatedpoints. Thisassumption
may seem unnecessary as all basic notions can be given in a perfect-free context.
However the preceding assumption is required for the development of some parts
of the theory. For example it is not even clear that the inequality
Σ1 (cid:2)X \Π1 (cid:2)X (cid:6)=∅
1 1
holds for every recursively presented metric space X, and although we will prove
this correct, it is still an open problem whether
Σ1 (cid:2)X \Π1 (cid:2)X (cid:6)=∅
1 (cid:2) 1
holds, if X is uncountable non-perfect.
Atfirstsighttheprecedingassumptionseemstohavenoimpacttotheapplica-
tions of effective theory to classical theory, at least from the level of Borel sets and
above, since every uncountable Polish space is Borel isomorphic to a perfect Polish
space, namely the Baire space. One has to be careful though, for it may be the
case that uncountably many spaces are involved, for example the sections of some
closedset. Inthis caseone would needtocollecttheprecedingBorelisomorphisms
in a uniform way, which may not be so easy to achieve.
Effectivetheoryhas alsofound itsway to“mixed-type”results, whereclassical
andeffectivenotionscoexistinthestatement,seeforexample[Lou80]and[Deb87]
for some very interesting results. In this type of results one should also take care
not to apply any theorems of effective theory which use the assumption about
perfectnesswithoutthespacebeingso. Thisisoftenself-evident(asitisforexample
in[Lou80]and[Deb87]),butneverthelessitwouldbegoodifonemakessurethat
this issue will not raise any concerns in future research on the area.
For these reasons it seemed useful to identify the main tools of effective theory
andthentracebacktheirproofstoseeiftheassumptionaboutperfectnessisindeed
1