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1. Descriptive Set Theory - Wikipedia, The Free Encyclopedia
In mathematical logic, Descriptive set theory is the study of certain classes of wellbehaved sets of real numbers, e.g. Borel sets, analytic sets,
http://en.wikipedia.org/wiki/Descriptive_set_theory
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Descriptive set theory
From Wikipedia, the free encyclopedia
Jump to: navigation search In mathematical logic descriptive set theory is the study of certain classes of " well-behaved sets of real numbers , e.g. Borel sets analytic sets , and projective sets . A major aim of descriptive set theory is to describe all of the "naturally occurring" sets of real numbers by using various constructions to build a strict hierarchy beginning with the open sets generated by the open intervals More generally, Polish spaces are studied in descriptive set theory; as it turns out, every Polish space is homeomorphic to a subspace of the Hilbert cube Many questions in descriptive set theory ultimately depend upon set-theoretic considerations and the properties of ordinal and cardinal numbers
edit See also
edit References
  • Kechris, Alexander S. (1994). Classical Descriptive Set Theory . Springer-Verlag. ISBN 0-387-94374-9 Moschovakis, Yiannis N. (1980). Descriptive Set Theory . North Holland. ISBN 0-444-70199-0

This mathematical logic -related article is a stub . You can help Wikipedia by expanding it Retrieved from " http://en.wikipedia.org/wiki/Descriptive_set_theory

2. Math 512 Descriptive Set Theory
In Descriptive set theory we try to avoid these pathologies by concentrating on natural classes of wellbehaved sets of reals, like Borel sets or projective
http://www.math.uic.edu/~marker/math512/
Math 512: Descriptive Set Theory
Fall 2002
MWF 11:00 216 Taft Hall
Instructor David Marker
  • Office: 411 SEO
  • Office Phone: (312) 996-3069
  • Office Hours: M 9-11, W 12-1 and by appointment
  • Fax: (312) 996-1491
  • e-mail: marker@math.uic.edu
  • web page: http://www.math.uic.edu/~marker
  • course web page: http://www.math.uic.edu/~marker/math512
Description
It is well know that when one studies arbitrary subsets of the real numbers one runs into many pathologies (non-measurable sets) and independent problems (the Continuum Hypothesis). In Descriptive Set Theory we try to avoid these pathologies by concentrating on natural classes of well-behaved sets of reals, like Borel sets or projective sets (the smallest class of sets containing Borel sets and closed under projections from higher dimenional spaces). While this is a restricted class of sets it includes most of the sets that arise naturally in mathematical practice. Lately there have been many intersting connections with dynamical systems, through the study of orbit equivalence relations.
The first half of the course will be devoted to developing the fundamental results and techniques of descriptive set theory. In the second half we will look at more recent developments. The exact topics covered will depend on the background and interest of the class.

3. Set Theory (Stanford Encyclopedia Of Philosophy)
Descriptive set theory traces its origins to the theory of integration by Henri Lebesgue The branch of Descriptive set theory known as Determinateness,
http://plato.stanford.edu/entries/set-theory/
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Set Theory
First published Thu 11 Jul, 2002
1. The Essence of Set Theory
The objects of study of Set Theory are sets . As sets are fundamental objects that can be used to define all other concepts in mathematics, they are not defined in terms of more fundamental concepts. Rather, sets are introduced either informally, and are understood as something self-evident, or, as is now standard in modern mathematics, axiomatically, and their properties are postulated by the appropriate formal axioms. The language of set theory is based on a single fundamental relation, called membership . We say that A is a member of B (in symbols A B ), or that the set B contains A as its element. The understanding is that a set is determined by its elements; in other words, two sets are deemed equal if they have exactly the same elements. In practice, one considers sets of numbers, sets of points, sets of functions, sets of some other sets and so on. In theory, it is not necessary to distinguish between objects that are members and objects that contain members the only objects one needs for the theory are sets. See the supplement Basic Set Theory for further discussion.

4. CLASSICAL DESCRIPTIVE SET THEORY
Descriptive set theory is one of the main and most active areas of research in presentday set theory, having strong connections both with topology and with
http://kurt.scitec.kobe-u.ac.jp/~brendle/descriptive.html
CLASSICAL DESCRIPTIVE SET THEORY
2002-2003, Spring Semester
Descriptive set theory is one of the main and most active areas of research in present-day set theory, having strong connections both with topology and with mathematical logic (in particular, recursion theory), as well as applications in areas as distinct as combinatorics, functional analysis, group theory etc... The subject of descriptive set theory is the structural investigation of definable (= easily describable) subsets of the real numbers and, more generally, of Polish (= separable completely metrizable) spaces. For example, from the topological point of view, the simplest sets of reals are the open sets and the closed sets. Next come the G (delta) sets (= countable intersections of open sets) and the F (sigma) sets (= countable unions of closed sets). Continuing this way, one builds up the family of Borel sets. Sets which are obtained as continuous images of Borel sets are called analytic, while their complements are coanalytic. Continuing with taking continuous images and complements, one constructs the

5. Fields Institute - Workshop On Descriptive Set Theory, Analysis And Dynamical Sy
Descriptive set theory is the study of definable sets and functions in Polish (complete separable metric) spaces. During the last few decades,
http://www.fields.utoronto.ca/programs/scientific/02-03/set_theory/workshop1/
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THEMATIC PROGRAMS
December 24, 2007
Thematic Program on Set Theory and Analysis
Workshop on Descriptive Set Theory, Analysis and Dynamical Systems
October 6-12, 2002
Organizers:
I. Farah (Staten Island College)
G. Hjorth (California, Los Angeles)
A.S. Kechris California Institute of Technology
Schedule and Abstracts Audio and Slides Thematic Year Homepage Housing and Hotels ... Visitor Information
Rationale:
Descriptive set theory is the study of definable sets and functions in Polish (complete separable metric) spaces. During the last few decades, intriguing connections have been discovered between descriptive set theory and other fields of mathematics, such as classical real analysis, probability theory, potential theory, harmonic analysis, Banach space theory, and ergodic theory.
More recently, a very promising new area, that is now very actively investigated, deals with the development of a theory of complexity of classification problems in mathematics, and the closely related theory of descriptive dynamics, i.e., the theory of definable actions of Polish groups on Polish spaces. A classification problem is in general the question of cataloging a class of mathematical objects up to some notion of equivalence by invariants.
This work brings descriptive set theory into contact with current developments in various areas of mathematics such as dynamical systems, including ergodic theory and topological dynamics, the theory of topological groups and their representations, operator algebras, geometric and combinatorial group theory, etc. Moreover, it provides new insights in the traditional relationships of descriptive set theory with other areas of mathematical logic, as, for example, with recursion theory, concerning the global structure of Turing degrees or with model theory, through the Topological Vaught Conjecture and the general study of the isomorphism relation on countable structures. This subject is now developing very rapidly but many fundamental questions still remain unanswered.

6. HAL :: [hal-00175696, Version 1] Omega-powers And Descriptive Set Theory
We study the sets of the infinite sentences constructible with a dictionary over a finite alphabet, from the viewpoint of Descriptive set theory.
http://hal.archives-ouvertes.fr/hal-00175696/en/

7. The Future Of Set Theory By S.Shelah
This naturally improves my view of parts of Descriptive set theory; as do the works on the number of equivalence classes (HrSh 152, Sh 202,) and on
http://shelah.logic.at/E16/E16.html
Abstract:
Judah has asked me to speak on the future of set theory, so as the next millennium is coming, to speak on set theory in the next millennium. But we soon cut this down to set theory in the next century. Later on I thought I had better cut it down to dealing with the next decade, but I suspect I will speak on what I hope to try to prove next year, or worse - what I have done in the last year (or twenty). It seems I am not particularly suitable for such a lecture, as I have repeatedly preferred to try to prove another theorem than to prepare the lecture (or article); so why did I agree at all to such a doubtful endeavor? Well, under the hypothesis that I had some moral obligation to help Haim in the conference (and the proceedings) and you should not let a friend down, had I been given the choice to help with organizing the dormitories, writing a lengthy well written expository paper or risking making fool of myself in such a lecture, I definitely prefer the latter.
The Future of Set Theory
Saharon Shelah
Institute of Mathematics
The Hebrew University of Jerusalem
91904 Jerusalem, Israel

8. Descriptive Set Theory -- From Wolfram MathWorld
Becker, H. and Kechris, A. S. The Descriptive set theory of Polish Group Actions. New York Cambridge University Press, 1996. CITE THIS AS
http://mathworld.wolfram.com/DescriptiveSetTheory.html
Search Site Algebra
Applied Mathematics
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Descriptive Set Theory The study of definable sets and functions in polish spaces SEE ALSO: [Pages Linking Here] REFERENCES: Becker, H. and Kechris, A. S. The Descriptive Set Theory of Polish Group Actions. New York: Cambridge University Press, 1996.
CITE THIS AS:
Weisstein, Eric W. "Descriptive Set Theory." From MathWorld A Wolfram Web Resource. http://mathworld.wolfram.com/DescriptiveSetTheory.html
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9. BRICS Mini-Course: A Taster Of Descriptive Set Theory
Descriptive set theory came into being around the turn of the (last!) century, as a means of analysing the ``difficulty of sets of real numbers,
http://www.brics.dk/MC/99/DescriptiveSet/index.html
BRICS Contents Programme References
A Taster of Descriptive Set Theory
A BRICS Mini-Course
September 9, 14 and 16, 1999 Lectures by
Julian Bradfield jcb@daimi.au.dk
BRICS and LFCS, Edinburgh
Course Contents
Descriptive set theory came into being around the turn of the (last!) century, as a means of analysing the ``difficulty'' of sets of real numbers, of the kind encountered in analysis. It is so called because it uses the logical description of a set as a measure of complexity. A few decades later, when recursion theory had been invented, similar ideas were used to describe sets of integers: the stuff of computation. It turned out that this ``effective'' descriptive theory and the older classical descriptive set theory are best seen as one subject; the combination brought great advances in both. The basic terminology of effective descriptive set theory pops up quite often in theoretical computer science: Pi -complete, for example. The classical notions are also seen, particularly when talking about timed systems and other things involving the reals. It is therefore often useful to have a nodding acquaintance with the terms and concepts of descriptive set theory. Furthermore, there are occasions when results of descriptive set theory have been used to surprising effect in TCS problems. However, it is also the case that descriptive set theory is a fascinating and beautiful subject, with surprising links to deep foundational issues in mathematics.

10. Descriptive Set Theory - Elsevier
Now available in paperback, this monograph is a selfcontained exposition of the main results and methods of Descriptive set theory.
http://www.elsevier.com/wps/product/librarians/501979
Home Site map Elsevier websites Alerts ... Descriptive Set Theory Book information Product description Author information and services Ordering information Bibliographic and ordering information Conditions of sale Book-related information Submit your book proposal Other books in same subject area About Elsevier Select your view DESCRIPTIVE SET THEORY
By
Y.N. Moschovakis
Included in series
Studies in Logic and the Foundations of Mathematics, 100
Description
Now available in paperback, this monograph is a self-contained exposition of the main results and methods of descriptive set theory. It develops all the necessary background material from logic and recursion theory, and treats both classical descriptive set theory and the effective theory developed by logicians.
Contents
The Basic Classical Notions. κ-Suslin and λ-Borel. Basic Notions of the Effective Theory. Structure Theory for Pointclasses. The Constructible Universe. The Playful Universe. The Recursion Theorem. Metamathematics. References. Index.
Paperback, ISBN: 0-444-70199-0, xii + 638 pages, publication date: 1980
Imprint: NORTH-HOLLAND Books and book related electronic products are priced in US dollars (USD), euro (EUR), and Great Britain Pounds (GBP). USD prices apply to the Americas and Asia Pacific. EUR prices apply in Europe and the Middle East. GBP prices apply to the UK and all other countries.

11. Analytic Sets In Descriptive Set Theory And NP Sets In Complexity Theory
20 20 C.Sureson, Descriptive set theory and Boolean Complexity theory , C.R.Acad.Sci.Paris, t.326 Série I, 1998, pages 255260.
http://portal.acm.org/citation.cfm?id=590140

12. The Descriptive Set Theory Of Polish Group Actions - Cambridge University Press
Descriptive set theory; 1. Polish groups; 2. Actions of polish groups; 3. Equivalence relations; 4. Invariant measures and paradoxical decompositions; 5.
http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=9780521576055

13. On Some Classical Problems Of Descriptive Set Theory
On some classical problems of Descriptive set theory. Vladimir Grigor evich Kanovei, Vasilii Aleksandrovich Lyubetskii Russian Mathematical Surveys 5855,
http://www.turpion.org/php/paper.phtml?journal_id=rm&paper_id=666

14. 03E: Set Theory
Descriptive set theory deals with the topological and measuretheoretic properties of the real line and related spaces. That is, beginning with the
http://www.math.niu.edu/~rusin/known-math/index/03EXX.html
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03E: Set theory
Introduction
Naive set theory considers elementary properties of the union and intersection operators Venn diagrams, the DeMorgan laws, elementary counting techniques such as the inclusion-exclusion principle, partially ordered sets, and so on. This is perhaps as much of set theory as the typical mathematician uses. Indeed, one may "construct" the natural numbers, real numbers, and so on in this framework. However, situations such as Russell's paradox show that some care must be taken to define what, precisely, is a set. However, results in mathematical logic imply it is impossible to determine whether or not these axioms are consistent using only proofs expressed in this language. Assuming they are indeed consistent, there are also statements whose truth or falsity cannot be determined from them. These statements (or their negations!) can be taken as axioms for set theory as well. For example, Cohen's technique of forcing showed that the Axiom of Choice is independent of the other axioms of ZF. (That axiom states that for every collection of nonempty sets, there is a set containing one element from each set in the collection.) This axiom is equivalent to a number of other statements (e.g. Zorn's Lemma) whose assumption allows the proof of surprising even paradoxical results such as the Banach-Tarski sphere decomposition. Thus, some authors are careful to distinguish results which depend on this or other non-ZF axioms; most assume it (that is, they work in ZFC Set Theory).

15. Ki, Haseo (1995-03-15) Topics In Descriptive Set Theory Related To Number Theory
Based on the point of view of Descriptive set theory, we have investigated several definable sets from number theory and analysis.
http://etd.caltech.edu/etd/available/etd-10112007-111738/
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Ki, Haseo (1995-03-15) Topics in descriptive set theory related to number theory and analysis. http://resolver.caltech.edu/CaltechETD:etd-10112007-111738
Type of Document Dissertation Author Ki, Haseo URN etd-10112007-111738 Persistent URL http://resolver.caltech.edu/CaltechETD:etd-10112007-111738 Title Topics in descriptive set theory related to number theory and analysis Degree PhD Option Mathematics Advisory Committee Advisor Name Title A. S. Kechris Committee Chair Keywords
  • none
Date of Defense Availability restricted Abstract NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. Based on the point of view of descriptive set theory, we have investigated several definable sets from number theory and analysis. In Chapter 1 we solve two problems due to Kechris about sets arising in number theory, provide an example of a somewhat natural [...] set, and exhibit an exact relationship between the Borel class of a nonempty subset X of the unit interval and the class of subsets of N whose densities lie in X. In Chapter 2 we study the A, S, T and U-sets from Mahler's classification of complex numbers. We are able to prove that U and T are [...]-complete and [...]-complete respectively. In particular, U provides a rare example of a natural [...]-complete set.

16. JSTOR New Directions In Descriptive Set Theory
June 1999 NEW DIRECTIONS IN Descriptive set theory ALEXANDER S. KECHRIS 1. I will start with a quick definition of Descriptive set theory It is the study
http://links.jstor.org/sici?sici=1079-8986(199906)5:2<161:NDIDST>2.0.CO;2-T

17. UIUC Dept. Of Mathematics Seminar Calendar
for Model theory and Descriptive set theory Seminar events the year of Friday, December 7, 2007. 1 day, 2 days, 1 week, 2 weeks, 1 month, 2 months, 3 months
http://torus.math.uiuc.edu/cal/math/cal?year=2007&month=12&day=07&interval=year&

18. IngentaConnect Topics In Invariant Descriptive Set Theory
The two concepts are elementary embeddability, from model theory, and analytic sets, from the usual Descriptive set theory.
http://www.ingentaconnect.com/content/els/01680072/2001/00000111/00000003/art000
var tcdacmd="dt";

19. SemProba: Le Résultat De Votre Recherche
22 matches for classification = Descriptive set theory Une démonstration du théorème de SouslinLusin (Descriptive set theory)
http://www-mathdoc.ujf-grenoble.fr/cgi-bin/sp?cla=Descriptive set theory

20. Books - Descriptive Set Theory And Forcing - 9781568811765
Buy Descriptive set theory and Forcing How to Prove Theorems About Borel sets the Hard Way - Price Range $52.60 - $53.35 from 2 sellers.
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21. Atlas: Applications Of Descriptive Set Theory To Cardinal Invariants By Jindrich
I will give a brief overview of the connection between Descriptive set theory and definable proper forcing and cardinal invariants, including the
http://atlas-conferences.com/c/a/k/f/02.htm
Atlas home Conferences Abstracts about Atlas Boise Extravaganza in Set Theory
March 28-30, 2003
Boise State University
Boise, ID, USA Organizers
Tomek Bartoszynski, Justin Moore View Abstracts
Conference Homepage Applications of descriptive set theory to cardinal invariants
by
Jindrich Zapletal
University of Florida I will give a brief overview of the connection between descriptive set theory and definable proper forcing and cardinal invariants, including the absoluteness theorems for Ciesielski-Pawlikowski type axioms, duality theorems, elimination of large cardinal assumptions using effective descriptive set theory, and some new examples of interesting forcing notions with non-set-theoretic motivation. I will present a new powerful preservation theorem for countable support iteration with descriptive set theoretic statement and proof. Date received: January 14, 2003 Atlas Conferences Inc. Document # cakf-02.

22. Descriptive Set Theory And Determinacy, Spring 2007
We cover chapters 11 and 33 of Jech s book set theory. Tapio Eerola will give the first set of lectures on chapter 11 and J.Kennedy will give the second
http://mathstat.helsinki.fi/kurssit/info/descriptive_set_theory_and_determinacy0
Department of Mathematics and Statistics
Faculty of Science
Faculty of Social Sciences
Departmental front page News ... People
Descriptive Set Theory and Determinacy, spring 2007
Lecturers
Tapio Eerola PhD Juliette Kennedy
Lectures
Weeks 3-9; Tuesdays 10-12 in room C321, Fridays 12-14 in room C123. First meeting on Jan. 16.
Contents
We cover chapters 11 and 33 of Jech's book "Set Theory." Tapio Eerola will give the first set of lectures on chapter 11 and J.Kennedy will give the second set of lectures on chapter 33. Students are invited to volunteer to relieve Tapio from time to time by lecturing in his place.
Credits
10 op, 5 ov

23. Springer Online Reference Works
Descriptive set theory was created in the early 20th century by the studies of . An important stage in the development of Descriptive set theory was the
http://eom.springer.de/d/d031380.htm

Encyclopaedia of Mathematics D
Article referred from Article refers to
Descriptive set theory The branch of set theory whose subject is the study of sets in dependence of those operations by which these sets may be constructed from relatively simple sets (e.g. closed or open subsets of a given Euclidean, metric or topological space). These operations include union, intersection, taking a complement or projection, etc. Descriptive set theory was created in the early 20th century by the studies of E. Borel R. Baire and H. Lebesgue in connection with the measurability of sets. Borel-measurable sets received the name of Borel sets or -sets (cf. Borel set ). On the other hand, Baire proposed a classification of functions, in so-called Baire function classes, and proved a number of theorems concerning these functions (cf. Baire classes Baire theorem ). Lebesgue showed that -sets are identical to Lebesgue sets of Baire functions (cf. Lebesgue set ), gave the first classification of -sets and showed that none of its classes was empty. The study of -sets became an important task of descriptive set theory, and the first such problem was the cardinality of

24. Bookpool: Classical Descriptive Set Theory
Classical Descriptive set theory. Alexander Kechris Springer, Hardcover, Published January 2005, 428 pages, ISBN 0387943749. List Price $69.95
http://www.bookpool.com/sm/0387943749
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25. Buy.com - Classical Descriptive Set Theory : ISBN 9780387943749
Classical Descriptive set theory ISBN 9780387943749 Book.
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Pricing Additional Info FREE SHIPPING Our Price: Shipping FREE Buy.com Total Price: Qty In Stock: Usually Ships in 1 to 2 business days. Format: Hardcover See all 4 New from What's this? Format: Hardcover ISBN: Publish Date: Publisher: Springer-Verlag TELOS Dimensions (in Inches) 9.75H x 6.5L x 1.25T Buy.com Sku: More about this product Item#: Buy.com Sales Rank: View similar products Product Summary Reviews Descriptive set theory has been one of the main areas of research in set theory for almost a century. This text attempts to present a largely balanced approach, which combines many elements of the different traditions of the subject. It includes a wide variety of examples, exercises (over 400), and applications, in order to illustrate the general concepts and results of the theory.
This text provides a first basic course in classical descriptive set theory and covers material with which mathematicians interested in the subject for its own sake or those that wish to use it in their field should be familiar. Over the years, researchers in diverse areas of mathematics, such as logic and set theory, analysis, topology, probability theory, etc., have brought to the subject of descriptive set theory their own intuitions, concepts, terminology and notation.

26. Book Classical Descriptive Set Theory (graduate Texts In Mathematics, Vol 156),
book undergraduate level (part ii) engineering colleges Descriptive set theory has been one of the main areas of research in set theory for almost a
http://www.lavoisier.fr/notice/gb035211.html
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Classical descriptive set theory (Graduate texts in mathematics, vol 156) Author(s) : KECHRIS
Publication date : 12-1994
Language : ENGLISH
Status : In Print (Delivery time : 10 days)
Comment Descriptive set theory has been one of the main areas of research in set theory for almost a century. This text attempts to present a largely balanced approach, which combines many elements of the different traditions of the subject. It includes a wide variety of examples, exercises (over 400), and applications, in order to illustrate the general concepts and results of the theory.
Subject areas covered:
  • Mathematics and physics Algebra analysis geometry Undergraduate level (part ii) - engineering colleges
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27. Dual Easy Uniformization And Model-Theoretic Descriptive Set Theory
It is well known that, in the terminology of Moschovakis, Descriptive set theory (1980), every adequate normed pointclass closed under $\forall^\omega$ has
http://projecteuclid.org/handle/euclid.jsl/1183743816
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28. ScienceDirect - Chaos, Solitons & Fractals : Quantum Gravity From Descriptive Se
In Descriptive set theory and the theory of polish spaces it is shown that 5. Definition 1. . ( ) theory and Descriptive set theory. This relation is
http://linkinghub.elsevier.com/retrieve/pii/S0960077903004533
Athens/Institution Login Not Registered? User Name: Password: Remember me on this computer Forgotten password? Home Browse My Settings ... Help Quick Search Title, abstract, keywords Author e.g. j s smith Journal/book title Volume Issue Page
Volume 19, Issue 5 , March 2004, Pages 1339-1344
Abstract Full Text + Links PDF (116 K) Related Articles in ScienceDirect Quantum gravity, Clifford algebras, fuzzy set theory an...
Quantum gravity, Clifford algebras, fuzzy set theory and the fundamental constants of nature
Volume 20, Issue 3 May 2004 Pages 437-450
M. S. El Naschie
Abstract
Subsequently we introduced a mathematical description of a transfinite, non-Archimedean geometry using descriptive set theory where a similar conclusion regarding the transfiniteness of quantum spacetime may be drawn from the existence of the Unruh temperature. In particular we introduced a straight forward logarithmic gauge transformation linking, as far as we are aware for the first time, classical gravity with the electroweak via a version of informational entropy. That way we found using and complexity theory that where is the dimensionless Newton gravity constant and is the fine structure constant at the electroweak unification scale.

29. Mathematical Logic Research - UC Berkeley Mathematics
John R. Steel, set theory, Descriptive set theory, fine structure. W. Hugh Woodin, set theory, large cardinals, mathematical logic.
http://math.berkeley.edu/research/logic.html
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Mathematical Logic Research
Mathematical Logic Research, UC Berkeley
Including model theory, recursion theory, and set theory
We have a large active group of researchers in several core areas of mathematical logic; including model theory, recursion theory and set theory. A number of members of the logic group belong to the Group in Logic and Methodology of Science, which runs a bi-weekly colloquium and has its own graduate students.
Faculty
Tenured and tenure-track:
John W. Addison (Emeritus), Theory of definability, descriptive set theory, model theory, recursive function theory.
Leo A. Harrington , Recursion theory, model theory, set theory.
Ralph N. McKenzie (Emeritus), General algebra, mathematical logic.
Thomas Scanlon , Model theory and applications to number theory.
Jack H. Silver , Mathematical logic, theory of sets.
Theordore A. Slaman , Recursion theory, mathematical logic.
Robert Solovay (Emeritus), Computer verification of formal proofs.

30. 1st European Set Theory Meeting In Będlewo, July 9 - 13, 2007
Descriptive set theory We intend to cover recent exciting progres on classification problems and definable equivalence relations, the work of Kechris,
http://www.logique.jussieu.fr/~boban/bedlewo/
1st European Set Theory Meeting Home Registration Participants ... Contact
Description and aim
Organized by B. Loewe G. Plebanek J. Vaananen B. Velickovic Set theory grew out of mathematical analysis through Georg Cantor 's work on sets of uniquness of trigonometric series in the late 19 th century. Over the last century it has developed into a vibrant and important subject of its own. On one hand it deals with questions of deep foundational impotance, such as the choice of axioms for mathematics and the questions of relative consistency of mathematical theories. On the other hand, techniques of combinatorial set theory can be applied successfully in a number of different areas of mathematics such as: general topology, measure theory, Banach space theory, abstract algebra, while descriptive set theory is applied in ergodic theory, dinamical systems, group representation theory, etc.
In the early days of set theory most of the advances were made in Europe. In particular, Polish mathematicians such as Sierpinski Kuratowski Mostowski and others played a key role in the early development of the subject. Among other things, they created the journal

31. CiNii - Some Results In The Effective Descriptive Set Theory : Dedicated To Prof
Some results in the effective Descriptive set theory Dedicated to Professor Motokiti Kondo on his sixtieth birthday anniversary
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Some results in the effective descriptive set theory : Dedicated to Professor Motokiti Kondo on his sixtieth birthday anniversary
TANAKA Hisao
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32. CAT.INIST
This is a survey of results in Descriptive set theory for domains and similar spaces, with the emphasis on the coalgebraic domains.
http://cat.inist.fr/?aModele=afficheN&cpsidt=18288798

33. Classical Descriptive Set Theory Is Available From Bestprices.com Books!
Classical Descriptive set theory only $72.68, get the Classical Descriptive set theory book From BestPrices.com!
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34. Award#0701030 - Applications Of Descriptive Set Theory In Ergodic Theory And Inv
Award Abstract 0701030 Applications of Descriptive set theory in Ergodic theory and investigations into singular cardinals combinatorics
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35. Uspekhi Matematicheskikh Nauk
On some classical problems of Descriptive set theory of some classical problems in Descriptive set theory which were formulated by Luzin Lusin and,
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UMN, 2003, Volume 58 Issue 5(353) (Mi umn666) This paper is cited in scientific articles by authors
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On some classical problems of descriptive set theory
V. G. Kanovei V. A. Lyubetskii
Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow
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UDC: Received: Citation: V. G. Kanovei, V. A. Lyubetskii, On some classical problems of descriptive set theory, UMN, 2003, Linking options:
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  • http://mi.mathnet.ru/eng/umn/58/5/3 Full text (in Russian): PDF file (1090 kB) References (in Russian): PDF file HTML file English version: Russian Mathematical Surveys, 2003, Review databases: http://www.ams.org/mathscinet-getitem?mr=2035719 http://www.zentralblatt-math.org/zmath/search/?an=1064.03031 Citing articles on Google Scholar: Russian citations English citations Related articles on Google Scholar: Russian articles English articles This paper is cited in the following Math-Net.Ru publications:

    • 36. Chapters.indigo.ca: Classical Descriptive Set Theory: A. S. Kechris: Books
    Descriptive set theory is the area of mathematics concerned with the study of the structure of definable sets in Polish spaces. Beyond being a central part
    http://www.chapters.indigo.ca/books/Classical-Descriptive-Set-Theory-A-S-Kechris
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    Classical Descriptive Set Theory
    Author: A. S. Kechris See more titles by A. S. Kechris Our Price:
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    Format: Hardcover Published: January 1, 1995 Dimensions: 402 Pages, 6.32 x 9.54 x 0.97 IN Published By: Springer Publishing Co., Inc.

    37. Mathematics Bonn -- Peter Koepke
    Descriptive set theory and infinitary games representation of sets of reals by systems of models of set theory, an alternative proof of the MartinSteel
    http://www.mathematics.uni-bonn.de/faculty/Koepke/
    Uni Bonn Mathematics Faculty studying at bonn ... contact * May 31, 1954 projects lectures seminars phd students ... homepage
    Peter Koepke
    Mathematical Institute
    Academic career
    Diplom, Bonn University Master of Arts, University of California, Berkeley PhD, Freiburg University Feodor-Lynen-Fellowship, Alexander von Humboldt Foundation, and Junior Research Fellow, Wolfson College, Oxford, England C1 assistant, Habilitation, Freiburg University C3 professor, Bonn University
    Invited lectures
    Logic in Hungary, Budapest, 2005
    Geometric Topology, Discrete Geometry and Set Theory (Keldysh Centennial), Moscow, 2004
    8th International Set Theory Workshop, Luminy, 2003
    New York City Logic Conference, New York, 2002
    Editorship
    Mathematical Logic Quarterly
    Research projects and activities     Determinacy axioms, infinitary combinatorics and their interactions (DFG project 2003-2006)
    Research profile
    Axiomatic set theory: determination of consistency strengths of infinitary combinatorial principles, using forcing and core models [ ]; characterizations of large cardinal axioms by embeddings of models of set theory [

    38. Descriptive Set Theory - Indopedia, The Indological Knowledgebase
    In mathematics, Descriptive set theory is the study of certain classes of wellbehaved sets of real numbers, eg Borel sets, analytic sets, and projective
    http://www.indopedia.org/Descriptive_set_theory.html
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    Categories     Set theory
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    Descriptive set theory
    ज्ञानकोश: - The Indological Knowledgebase In mathematics descriptive set theory is the study of certain classes of " well-behaved sets of real numbers , e.g. Borel sets analytic sets , and projective sets . A major aim of descriptive set theory is to describe all of the "naturally occurring" sets of real numbers by using various constructions to build a strict hierarchy beginning with the open sets generated by the open intervals More generally, Polish spaces are studied in descriptive set theory; as it turns out, every Polish space is homeomorphic to a subspace of the Hilbert cube Many questions in descriptive set theory ultimately depend upon set-theoretic considerations and the properties of ordinal and cardinal numbers edit
    References
    • A. Kechris, Classical Descriptive Set Theory, GTM 156, Springer-Verlag, 1995. Y. Moschovakis, Descriptive Set Theory, North-Holland, 1980.

    Retrieved from " http://www.indopedia.org/Descriptive_set_theory.html

    39. CiteULike: Tag Descriptive-set-theory [6 Articles]
    posted to Descriptiveset-theory by dmitri83 on 2006-10-11 152249 as. Geometry in Urysohn s universal metric space. by Julien Melleray
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  • The complexity of classifying separable Banach spaces up to isomorphism (9 Oct 2006) by Valentin Ferenczi , Alain Louveau , Christian Rosendal posted to descriptive-set-theory by on 2006-10-11 15:22:49 as Compact metrizable groups are isometry groups of compact metric spaces (24 May 2005) by Julien Melleray posted to descriptive-set-theory by on 2006-10-11 17:41:24 as along with 1 person ansobol Computing the complexity of the relation of isometry between separable Banach spaces (17 Nov 2005) by Julien Melleray posted to descriptive-set-theory by on 2006-10-11 17:44:13 as along with 1 person TooMuchCoffeeMan Automatic continuity of homomorphisms and fixed points on metric compacta (26 Apr 2006) by Christian Rosendal , Slawomir Solecki posted to descriptive-set-theory by on 2006-10-24 14:23:35 as Geometry in Urysohn's universal metric space (24 May 2005) by Julien Melleray posted to descriptive-set-theory by on 2006-10-11 17:24:35 as along with 1 person ansobol Some isometry groups of the Urysohn space Annals of Pure and Applied Logic , Vol. 143, No. 1-3. (November 2006), pp. 70-78.

    • 40. [FOM] Polish Spaces And Descriptive Set Theory
    Original Message From Jan Pax pax0 at seznam.cz Can someone please give me the reason why in Descriptive set theory we prefer to work with
    http://cs.nyu.edu/pipermail/fom/2007-September/011904.html
    [FOM] Polish spaces and descriptive set theory
    joeshipman@aol.com joeshipman at aol.com
    Fri Sep 7 16:23:45 EDT 2007 pax0 at seznam.cz > Can someone please give me the reason why in descriptive set theory we prefer to work with perfect Polish spaces (i.e. toplogical spaces homeomorphic to a complete separable metric space without isolated points) like Baire space w^w, instead of the more intuitive real line? The real line is also a complete separable metric space without isolated points, but it has properties which, while useful for analysis, are irrelevant for measure theory and logic. Baire space has the property that it is homeomorphic to its own square, which greatly simplifies the study of the kinds of phenomena descriptive set theorists care about (while collapsing and trivializing phemomena like "dimension" which they don't care so much about). JS Email and AIM finally together. You've gotta check out free AOL Mail! - http://mail.aol.com

    41. 80-611 Computability And Incompleteness Instructor Jeremy Avigad
    Descriptive set......21624 Topics in analysis classical Descriptive set theory Instructor Ernest Schimmerling MW 330-450 PH 126A 12 Units
    http://logic.cmu.edu/pal-courses-s07.txt
    80-611 Computability and Incompleteness Instructor: Jeremy Avigad Units: 12 TR 10:30AM - 11:50AM BH A51 Description: The 1930's witnessed two revolutionary developments in mathematical logic: first, Goedel's famous incompleteness theorems, which demonstrate the limitations of formal mathematical reasoning, and second, the formal analysis of the notion of computation in the work of Turing, Goedel, Herbrand, Church, Post, Kleene, and others, together with Turing's results on the limits of computation. This course will cover these developments, and related results in logic and the theory of computability. 80-315/80-615 Modal Logic Instructor Horacio Arl³-Costa Units: 9-12 Units TR: 03:00PM-04:20PM, BH 150 Description: An introduction to first-order modal logic. The course considers several modalities aside from the so-called alethic ones (necessity, possibility). Epistemic, temporal or deontic modalities are studied, as well as computationally motivated modals (like "after the computation terminates"). Several conceptual problems in formal ontology that motivated the field are reviewed, as well as more recent applications in computer science and linguistics. Special attention is devoted to Scott-Montague models of the so-called 'classical' modalities. Incompleteness results are reviewed and a general completeness result for all classical systems is presented in terms of first order general neighborhood frames. 21-624 Topics in analysis: classical descriptive set theory Instructor: Ernest Schimmerling MW 3:30-4:50 PH 126A 12 Units Description: Descriptive set theory combines analysis and logic. The required background is undergraduate level analysis, logic and set theory. The recommended textbook is Kechris, "Classical Descriptive Set Theory", Graduate Texts in Mathematics 156, Springer-Verlag. Another useful book is Moschovakis, "Descriptive Set Theory" (out of print). I posted a very general course description online at http://www.math.cmu.edu/~eschimme/21-624/syllabus.pdf The topics that will be covered include: Polish spaces (Baire space, Cantor space, etc.); classical definability hierarchies (Borel Hierarchy, Projective Hierarchy); effective definability hierarchies; the Wadge hierarchy; complete sets, universal sets; separation; reduction; uniformization; prewellordering; scales; perfect sets; Baire measurability; Lebesgue measurability; Suslin operation; substitution property; basis theorems; tree representations; proofs of determinacy (Borel determinacy); applications of determinacy; definable winning strategies; theorems on the boundedness of norms; theorems on the ranks of wellfounded relations (Kunen-Martin); admissible ordinals; stable ordinals 21-700 Mathematical Logic II Instructor: Peter Andrews MWF 12:30 PH 225B 12 units DESCRIPTION: This course is open to, and suitable for, students who have had an introductory logic course such as 21-300 Basic Logic, 21-600 Mathematical Logic I, or 80-310 Logic and Computation. After one has learned the basics of logic from such a course, it is natural to consider formal systems in which one can formalize mathematics and other disciplines. This requires the ability to quantify over variables for predicates, sets and functions as well as individuals; indeed, one needs to be able to discuss sets of sets, sets of sets of sets, etc., as well as functions whose arguments and values can be functions and sets of various types. Such a system, called type theory, was developed by the eminent philosopher Bertrand Russell, who used it as the logical basis for the great three-volume work Principia Mathematica (written jointly with Alfred North Whitehead), which provided substantial support for Russell's then novel thesis that all of mathematics is a branch of logic. 21-700 provides an introduction to a version of type theory due to Alonzo Church which contains lambda-notation for functions; it is bot