Online Store

Geometry.Net - the online learning center
Home  - Mathematical_Logic - Descriptive Set Theory
Page 1     1-59 of 59    1 

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,
var wgNotice = ""; var wgNoticeLocal = ""; var wgNoticeLang = "en"; var wgNoticeProject = "wikipedia";
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 "

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
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:
  • web page:
  • course web page:
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,
Cite this entry Search the SEP Advanced Search Tools ...
Please Read How You Can Help Keep the Encyclopedia Free
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.

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
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,
Home About Us NPCDS/PNSDC Mathematics Education ... Search
December 24, 2007
Thematic Program on Set Theory and Analysis
Workshop on Descriptive Set Theory, Analysis and Dynamical Systems
October 6-12, 2002
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
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.

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
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
Search Site Algebra
Applied Mathematics

Calculus and Analysis
... General Set Theory
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.
Weisstein, Eric W.
"Descriptive Set Theory." From MathWorld A Wolfram Web Resource.
Wolfram Research, Inc.
Other Wolfram Sites: Wolfram Research Demonstrations Site Integrator Tones Functions Site Wolfram Science Download Mathematica Player Complete Mathematica Show off your math savvy with a MathWorld T-shirt

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,
BRICS Contents Programme References
A Taster of Descriptive Set Theory
A BRICS Mini-Course
September 9, 14 and 16, 1999 Lectures by
Julian Bradfield
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.
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
Y.N. Moschovakis
Included in series

Studies in Logic and the Foundations of Mathematics, 100

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.
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.

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.

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,

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
Search Subject Index MathMap Tour ... Help! ABOUT: Introduction History Related areas Subfields
POINTERS: Texts Software Web links Selected topics here
03E: Set theory
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.
Caltech Library System
Browse Search Caltech Student Instructions
Ki, Haseo (1995-03-15) Topics in descriptive set theory related to number theory and analysis.
Type of Document Dissertation Author Ki, Haseo URN etd-10112007-111738 Persistent URL 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<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

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.
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) 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.
Go back to home page Login Register SHOP FOR IN All Products Appliances Auto Parts Books Cameras Clothing Computers Electronics Furniture Indoor Living Magazines Movies Music Musical Instruments Office Outdoor Living Software Sporting Goods Toys Video Games SEARCH Sell Yours Save Product to Your List(s) (Javascript required) Set Price Alert
Descriptive Set Theory and Forcing (English)
(How to Prove Theorems About Borel Sets the Hard Way - ISBN: 9781568811765) Price range: from 2 Sellers Publisher: A K Peters Ltd Format: Hardcover MSRP: $ 55.01 Synopsis: Not Available User Reviews Not Rated Write a Review New (2 Sellers from $52.60) Enter Zip Code* Seller Price (USD) Tax* Shipping* BottomLinePrice* Availability Seller Rating
Merchant Info
No Tax Free Your Best Price
In Stock
2323 Reviews
Merchant Info
No Tax In Stock
1542 Reviews
Enter Zip Code* * Enter your zip code above to get the best price for delivery in your area including shipping and tax. Shipping costs are based on an estimate of UPS ground or equivalent carrier within the contiguous US, excluding Alaska and Hawaii. Please see Seller's website for actual shipping costs. Do you see a mistake?

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
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
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
Department of Mathematics and Statistics
Faculty of Science

Faculty of Social Sciences

Departmental front page
News ... People
Descriptive Set Theory and Determinacy, spring 2007
Tapio Eerola PhD Juliette Kennedy
Weeks 3-9; Tuesdays 10-12 in room C321, Fridays 12-14 in room C123. First meeting on Jan. 16.
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.
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

Encyclopaedia of Mathematics
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
Classical Descriptive Set Theory Alexander Kechris
Springer, Hardcover, Published January 2005, 428 pages, ISBN 0387943749 List Price: $69.95
Our Price:
You Save: $17.00 (24% Off)
Availability: In-Stock Be the First to Write a Review and tell the world about this title! Books on similar topics, in best-seller order: Books from the same publisher, in best-seller order:
Forgot your password?

Shipping Options ... Log In

25. - Classical Descriptive Set Theory : ISBN 9780387943749
Classical Descriptive set theory ISBN 9780387943749 Book.
My Account Wishlist Help All Departments ... Warranties Books by Title by Author by ISBN by Publisher
Save $30 Instantly with the Visa Card. Click here. Books Best-sellers New ... Rewards UpdateProductViewHistoryCookie('Classical Descriptive Set Theory','30055946');
Classical Descriptive Set Theory (Hardcover)
Product Image enlarge image
Pricing Additional Info FREE SHIPPING Our Price: Shipping FREE 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 Sku: More about this product Item#: 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
Search on All Book CD-Rom eBook Software The french leading professional bookseller Description
Approximate price

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
New search
Your basket
Information New titles
E-books Customer services Open an account Ordering non-listed items Order tracking Help Back to the home page Company information Terms and conditions Partner's sites ... basket Special Offer

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
Log in RSS Title Author(s) Abstract Subject Keyword All Fields FullText more options

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
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
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
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.
Department of Mathematics Mathematics Department Home About Us People Courses ... Building Emergency Plan
Navigation Search Site Map FAQs
Google Services
Math WWW
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.
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,
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
Top Page Browse Publications Citation Index CiNii+Citation Index ... Japanese Journal Title
Publications of the Research Institute for Mathematical Sciences, Kyoto University. Ser. A
Vol.3, No.1(19670700) pp. 11-52 Kyoto University Bibliography
Some results in the effective descriptive set theory : Dedicated to Professor Motokiti Kondo on his sixtieth birthday anniversary
College of Engineering, Hosei University Read/Search Full Text Holdings NII Article ID (NAID) NII NACSIS-CAT ID (NCID) Text Lang ENG Databases NII-ELS Export Refer/BibIX Format BibTex Format Tab Separated Text (TSV) NII HOME ... NII-REO National Institute of Informatics

This is a survey of results in Descriptive set theory for domains and similar spaces, with the emphasis on the coalgebraic domains.

33. Classical Descriptive Set Theory Is Available From Books!
Classical Descriptive set theory only $72.68, get the Classical Descriptive set theory book From!

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

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,
... What is RSS
UMN: Year: Volume: Issue: Page: Find
Personal entry: Login: Password: Save password Enter Forgotten password? Register
UMN, 2003, Volume 58 Issue 5(353) (Mi umn666) This paper is cited in scientific articles by authors
PDF version
HTML version
On some classical problems of descriptive set theory
V. G. Kanovei
V. A. Lyubetskii
Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow

UDC: Received: Citation:
V. G. Kanovei, V. A. Lyubetskii, On some classical problems of descriptive set theory, UMN, 2003, Linking options:
  • Full text (in Russian): PDF file (1090 kB) References (in Russian): PDF file HTML file English version: Russian Mathematical Surveys, 2003, Review databases: 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. 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
    In Books In Books Outlet In Toy Store In DVD In Music In iPod Search All Where Canadians shop for books, DVDs, kid's toys, games and music CDs at Canada's online bookstore -
    Classical Descriptive Set Theory
    Author: A. S. Kechris See more titles by A. S. Kechris Our Price:
    Member Price:
    In Stock
    Rate this Item
    Average Customer Rating
    0 ratings
    Community Reviews
    be the first to write a review!
    About this Book
    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
    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
    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
    Indopedia Main Page FORUM Help ... Log in The Indology CMS
    Set theory
    Printable version
    Wikipedia Article
    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
    • A. Kechris, Classical Descriptive Set Theory, GTM 156, Springer-Verlag, 1995. Y. Moschovakis, Descriptive Set Theory, North-Holland, 1980.

    Retrieved from "

    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
    Register Log in FAQ
    Tag descriptive-set-theory [6 articles]
    Recent papers classified by the tag descriptive-set-theory.
  • 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 Can someone please give me the reason why in Descriptive set theory we prefer to work with
    [FOM] Polish spaces and descriptive set theory joeshipman at
    Fri Sep 7 16:23:45 EDT 2007 pax0 at > 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! -

    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
    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 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 both an enhancement and a simplification of Russell's type theory. Type theory is also known as higher-order logic, since it incorporates not only first-order logic, but also second-order logic, third-order logic, etc. The notation of this version of type theory is actually very close to that of traditional mathematics, and it has been found to be a good language for use in automated theorem proving systems which prove theorems of mathematics involving sets and functions. Another important application of type theory is in the formal specification and verification of hardware and software systems. Familiarity with Church's type theory provides fundamental background for the study of denotational semantics for functional programming languages. 21-700 starts with a general treatment of the syntax and semantics of type theory. Students use the computer program ETPS (Educational Theorem Proving System) as an aid in constructing certain formal proofs. Skolem's paradox about countable models for formalizations of set theory in which one can prove the existence of uncountable sets is resolved with the aid of the important distinction between standard and nonstandard models. It is shown that theories which have infinite models must have nonstandard models. Henkin's Completeness Theorem is proved. Attention then turns to showing how certain fundamental concepts of mathematics can be formalized in type theory. It is shown how cardinal numbers and the set of natural numbers can be defined, and Peano's Postulates are derived from an Axiom of Infinity. It is shown how recursive functions can be represented very elegantly in type theory. The last part of the course concerns the fundamental limitations of any system in which mathematics can be formalized. The famous incompleteness, undecidability and undefinability results of Godel and Tarski are presented, along with Lob's Theorem about the sentence which says "I am provable". The rich notation of type theory makes it possible to present very elegant proofs of these deep theorems. TEXT: Peter B. Andrews, An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof, Second Edition, Kluwer Academic Publishers, 2002, chapters 5-7. 21-703 Model Theory II Instructor: Rami Grossberg MWF 1:30-2:20PM PH A21A 12 Units Description: The subject started 30 years ago by Shelah, the goal is to discover the concepts and tools necessary for the development of model theory for infinatry logic and ultimately to have a complete theory of invariants of models up to isomorphism whenever this is possible. Shelah also proposed the a conjecture as a test for the development of the theory: Shelah's categoricity conjecture, it is a parallel to Morley's categoriicty theorem for Lw1,w. Despite the existence of about a thousand pages of partial results the conjecture is still open. More details can be found on the course webpage at:

    42. The Math Forum - Math Library - Set Theory
    Research in the group is concentrated on axiomatic set theory, in particular Inner models and large cardinals; Descriptive set theory and Determinacy;
    Browse and Search the Library
    Math Topics Logic/Foundations : Set Theory

    Library Home
    Search Full Table of Contents Suggest a Link ... Library Help
    Selected Sites (see also All Sites in this category
  • The Beginnings of Set Theory - MacTutor Math History Archives
    Linked essay describing the rise of set theory from Cantor (with discussion of earlier contributions) through the first half of the 20th century, with another web site and 25 references (books/articles). more>>
  • Interactive Basic Math Sets - Martin Selditch
    A tutorial on sets, convering the definition of sets and their elements, union, intersection, subsets, and sets of numbers. more>>
  • Set Theory - Dave Rusin; The Mathematical Atlas

    All Sites - 70 items found, showing 1 to 50
  • Around the Goedel's Theorem - Karlis Podnieks
    A draft translation of Podnieks' book, published in 1992 in Russian. Contents include: Platonism, intuition and the nature of mathematics; Axiomatic set theory; First order arithmetic; Hilbert's Tenth problem; Incompleteness theorems; Around the Goedel's ...more>>
  • Bell Package - Jacek Kisynski This package provides functions which are useful while dealing with set partitions. We provide (hopefully) fast methods for sets of size up to 15 and methods with no set size restrictions which use BigInteger objects. The later ones are constrained
  • 43. EULER Record Details
    Descriptive set theory and dynamical systems. Based on the international workshop, Foreman, M.CA International Workshop on Descriptive set theory and

    44. Wiki Tree (descriptive Set Theory)
    In Descriptive set theory, a tree on a set math X /math is a set of finite sequences of elements of math X /math that is closed under subsequences.
    Wiki: Tree (descriptive set theory) Auto insurance- free quote now (Ad) In descriptive set theory , a tree then In particular, every nonempty tree contains the empty sequence. A branch body A tree that has no branches is called wellfounded ; a tree with at least one branch is illfounded Home Licensing Wapedia: For Wikipedia on mobile phones

    45. Set Theory Papers Of Andreas R. Blass
    Baer Meets Baire Applications of Category Arguments and Descriptive set theory to Z^{aleph_0}, joint with John Irwin (Abelian Groups and Modules (ed.
    Set Theory Papers
    Andreas Blass
    The papers are listed in reverse chronological order, except that I put two surveys at the beginning to make them easier to find. Nearly Countable Cardinals PostScript or PDF An expository talk, for a general mathematical audience, about cardinal characteristics of the continuum. Combinatorial Cardinal Characteristics of the Continuum (to appear as a chapter in the Handbook of Set Theory (ed. M. Foreman, M. Magidor, and A. Kanamori)) PostScript or PDF This survey of the theory of cardinal characteristics of the continuum is to appear as a chapter in the "Handbook of Set Theory." As the title indicates, I concentrate on the combinatorial characteristics; Tomek Bartoszynski has written a chapter on the category and measure characteristics. Voting Rules for Infinite Sets and Boolean Algebras PDF A voting rule in a Boolean algebra B is an upward closed subset that contains, for each element x in B, exactly one of x and -x. We study several aspects of voting rules, with special attention to their relationship with ultrafilters. In particular, we study the set-theoretic hypothesis that all voting rules in the Boolean algebra of subsets of the natural numbers modulo finite sets are nearly ultrafilters. We define the notion of support of a voting rule and use it to describe voting rules that are, in a sense, as different as possible from ultrafilters. Finally, we consider how much of the axiom of choice is needed to guarantee the existence of voting rules.

    46. Descriptive Set Theory And The Structure Of Sets Of Uniqueness - Cambridge Unive
    In this book are developed the intriguing and surprising connections that the subject has with Descriptive set theory. These have only been discovered

    47. CRC Press Online
    With this active field of research receiving increased attention from various mathematical fields, Invariant Descriptive set theory provides a comprehensive

    48. ASL Info Main Page
    His doctoral dissertation was entitled Applications of Descriptive set theory to topology and analysis and was notable for its surprising results
    ASL Information About the ASL
    Contact Us


    Prizes and Awards

    Committees, Reports

    Member Search Sacks Prize Recipients 2006 Sacks Prize

    Matteo Viale, University of Torino and the University of Paris 7 Viale received his Ph.D. in 2006 from the University of Torino and the University of Paris 7, under the supervision of Alessandro Andretta and Boban Velickovic. The Committee's citation reads: "Viale's thesis makes fundamental contributions to our understanding of the consequences of forcing axioms in the combinatorics of singular cardinals. In particular, it solves a well-known problem, by showing that the Proper Forcing Axiom implies the Singular Cardinals Hypothesis.'' 2005 Sacks Prize
    Montalbán received his Ph.D. in 2005 from Cornell University, under the supervision of Richard Shore. The Committee on Prizes and Awards' citation reads: "The thesis, entitled Beyond the Arithmetic

    49. Intute Science, Engineering And Technology - Full Record Details For Descriptive
    Title, Descriptive set theory Math 512 Fall 2002 They discuss classical Descriptive set theory, Borel sets, the influence of recursion theory on

    50. – Discount Bookstore. Bestsellers, New Books, Used Books An
    This book is about actions of Polish groups, in connection withor from the point of view ofthe subject of Descriptive set theory. Descriptive set theory
    No Checkout My Account Help Loading...
    Free Shipping. $
    or more. Details here! .: HOME .: BESTSELLERS ... .: CLEARANCE BOOKS SEARCH Books Title Author ISBN Advanced Search Browse Art
    Item Detail The Descriptive Set Theory of Polish Group Actions Author(s): Howard Becker, Alexander S. Kechris Cover: Paperback New Copy: Usually Ships in 7-10 Business Days List Price Our Price You save $1.78 Synopsis: A Polish space (group) is a separable, completely metrizable topological space (group). This book is about actions of Polish groups, in connection withor from the point of view ofthe subject of descriptive set theory. Descriptive set theory is the study of definable sets and functions in Polish spaces. The basic classes of definable sets are the classes of Borel, analytic and coanalytic sets, and these constitute the main topic of the book, but the authors also consider other classes of definable sets. This will be a valuable book for all researchers in set theory and related areas. Check Out These Amazing Deals!

    51. Baztech Informacja O Publikacji
    Tytul Analytic sets in Descriptive set theory and NP sets in Complexity theory. Czasopismo Annales Societatis Mathematicae Polonae.

    52. Mathematics Under The Microscope: Explication Of Explicitness
    The concept of a Borel set belongs to the area of mathematics called Descriptive set theory. I have special feelings towards the Descriptive set theory and
    @import url(""); @import url(""); var BL_backlinkURL = "";var BL_blogId = "32727103";
    Mathematics under the Microscope
    Atomic objects, structures and concepts of mathematics
    Tuesday, September 12, 2006
    Explication of Explicitness
    The following informal concepts of mathematical practice cry out to be explicated: beautiful, natural, deep, trivial, "right", difficult, genuinely, explanatory ... Timothy Gowers
    These words by Tim Gowers deserve to be called "Gowers' programme". He formulated it in his talk at the conference Mathematical Knowledge in Cambridge in June 2004.
    This my post will demonstrate two extreme cases among many possible approaches to the explication of informal mathematical concepts: a lazy metamathematical approach and a hard-core mathematical treatment (terminology is mine). An example of a lazy metamathematical approach was given by Gowers in the same talk. He said that, in his opinion, a "comprehensible" proof (yet another frequently used informal concept) is not necessarily the shortest one, but a proof of small width . Here, "width" measures how much you must hold in your head at any one time. Alternatively, imagine that you write a detailed proof on a blackboard, carefully referring to all intermediate steps. However, if you know that a certain formula or lemma will never be used again, you erase it and re-use the space. A "small width" proof is a proof which never expands beyond one (small) blackboard. It is a nice naive concept which perhaps can be made into a rigorous theory. I do not know, however, whether proof theorists and complexity theorists have been sufficiently interested and if so, successful in its development.

    53. Leszek Pacholski: Papers
    260262; Leszek Pacholski, Book review Y. N. Moschovakis, Descriptive set theory, Studia Logica, 41, 1982, pp. 429-430; Leszek Pacholski,

    54. Keldysh Conference
    She first worked in Descriptive set theory (being a member of the celebrated Lusitania a mathematiciancommunity grouped around N.N. Lusin), and then,
    (Moscow, August 24 - 28, 2004) IN CELEBRATION OF THE CENTENNIAL
    of Ljudmila Vsevolodovna Keldysh
    (March 12, 1904 - February 16, 1976) Scopes and Themes The conference will cover the areas of L.V. Keldysh’s scientific interests as well as some modern development in geometry and topology. There will be four sections:
    • Geometric Topology (A.V. Chernavsky , E.V. Shchepin)
    • Set Theory (V.V. Fedorchouk, V.G. Kanovei, V.A. Lyubetsky)
    • Topology of Low Dimensions (S.V. Matveev, I.A. Dynnikov)
    • Combinatorial geometry and algebraic topology (V.M. Buchstaber)

    Ljudmila Vsevolodovna Keldysh was an eminent mathematician. She first worked in Descriptive Set Theory (being a member of the celebrated Lusitania - a mathematiciancommunity grouped around N.N. Lusin), and then, from the 50's, in the domain of Geometric Topology.
    Her main result in Descriptive Theory consists in the full analysis of the Borel sets, including the construction of arithmetic examples for all effectively given transfinites.

    Page 1     1-59 of 59    1