In fact, the classical cantor i set is a set of uniqueness. Descriptive set theory is the area of mathematics concerned with the study of the structure of definable sets in polish spaces. The first chapter consists of the study of the descriptive. Descriptive set theory has been one of the main areas of research in set theory for almost a century. On the development of descriptive set theory unt digital. The language of set theory can be used to define nearly all mathematical objects. Thearithmetichierarchy 27 4 analyticsets 34 5 coanalyticsets 43 6 determinacy 54 7 hyperarithmeticsets 62 ii borelequivalencerelations 73 8.
Descriptive set theory and model theory third lecture. It is much more interesting and relevant in this context, however, to ask whether there is actually a definable such correspondence. The axiom of choice, the lemma of zorn and the hausdor maximal principle 140 appendix. Christian rosendal descriptive set theory and model theory notre dame, june 2016 7 22 examples examples of fra ss e limits q. In other words, a perfect set is a closed set that has no isolated points. Christian rosendal descriptive set theory and model theory notre dame, june 2016 4 27 to see this, note that since s 0 is nite, there is some k so that every s 0 2s 0 can be written as a product. This textbook gives an introduction to axiomatic set theory and examines the prominent questions that are relevant in current research in a manner that is accessible to students. Thus effective descriptive set theory combines descriptive set theory with recursion theory. In mathematical logic, descriptive set theory dst is the study of certain classes of wellbehaved subsets of the real line and other polish spaces. Pdf this thesis deals with the descriptive set theory and the geometry of banach spaces. Reference request, descriptive set theory mathematics. Descriptive set theory and uncountable model theory michael c. Descriptive set theory davidmarker fall2002 contents i classicaldescriptivesettheory 2 1 polishspaces 2 2 borelsets 14 3 e. A to say that a is a member element of a, and we write a.
Notes on descriptive set theory and applications to banach spaces. The present book covers each of these areas, giving the reader an understanding of the ideas involved. A developing set of notes i have used in teaching 220abc, the basic graduate course in mathematical logic at ucla. This alone assures the subject of a place prominent in human culture.
But even more, set theory is the milieu in which mathematics takes place today. Examples include the real line, the baire space, the cantor space, and the hilbert cube. Sets fundamental to set theory is the notion of membership. Beyond being a central part of contemporary set theory, the concepts and results of descriptive set theory are being used in diverse fields of mathematics, such as logic, combinatorics, topology, banach space theory, real and harmonic analysis.
Function spaces provide other classical examples of polish spaces. Our aim has been to present some recent work in descriptive set theory and its applications to an area of harmonic analysis. New directions in descriptive set theory makes essential use of the axiom of choice in establishing a bijection with the reals. In descriptive set theory we mostly study polish spaces such as the baire space, the cantor space, and the reals. Kechris 3 coarse geometry of polish groups 4 geometry of automorphism groups christian rosendal descriptive set theory and model theory notre dame, june 2016 2 27. Set theory, with an introduction to descriptive set theory. Beyond being a central part of contemporary set theory, the concepts and results of descriptive set theory are being used in diverse fields of mathematics, such as logic, combinatorics, topology, banach space theory, real and harmonic analysis, potential theory. Heuristically, it is a complete separable metric space whose metric has been forgotten. Descriptive set theory is the study of definable subsets of polish spaces, where definable is taken to mean from the borel or projective hierarchies. It includes a wide variety of examples, exercises over 400, and. Other topics include infinite games and determinacy, definable equivalence relations and borel reductions between them, polish groups, and effective descriptive set theory. Typical uses of descriptive set theory in analysis are most often through regularity properties of definable sets, like measurability, the property of baire, capacitability, etc.
I will start with a quick definition of descriptive set theory. An introduction to classical descriptive set theory. Set theory has experienced a rapid development in recent years, with major advances in forcing, inner models, large cardinals and descriptive set theory. Descriptive set theory is the study of sets in separable, complete metric spaces that can be defined or constructed, and so can be expected to have special properties not enjoyed by arbitrary pointsets. Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. A preliminary version of the second greek edition of notes on set theory. Questions about the borel hierarchy, the projective hierarchy, polish spaces, infinite games and determinacy related topics, all fit into this category very well. Descriptive set theory university of illinois at chicago. This subject was started by the french analysts at the turn of the 20th century, most prominently lebesgue, and, initially, was concerned primarily with establishing regularity properties of. Dec 11, 2017 descriptive set theory was originally developed on polish spaces. As well as being one of the primary areas of research in set theory, it has applications to other areas of mathematics such as functional analysis, ergodic theory, the study of operator algebras and group actions, and mathematical logic. Yiannis n moschovakis this monograph develops descriptive set theory systematically, from its classical roots to the modern effective theory and the consequences of strong especially determinacy hypotheses.
Classical descriptive set theory alexander kechris springer. Publication date 1980 topics descriptive set theory. Topics covered include corel lattices, universal sets, the operation a, analytic sets, coanalytic sets, and the continuum hypothesis the appendix contains a translation of the. Descriptive set theory has been one of the main areas of research in set theory. Descriptive set theory and harmonic analysis cambridge core. Notes on axioms of set theory well ordering and ordinal numbers123 1. This thesis deals with the descriptive set theory and the geometry of banach spaces. Examples of equivalence relations and polish group actions. A subset of a topological space is borel if it is in the algebra generated by the open sets. Lecture notes on descriptive set theory jan reimann department of mathematics pennsylvania state university notation u x ball of radius about x u topological closure of u 2 x. Introduction to descriptive set theory department of mathematics. Obviously, r itself is perfect, as is any closed interval in r. In chapter 2, we define and understand the objects of study of descriptive set theory, the polish spaces.
The newer direction is the application of descriptive set theory toward other fields in mathematics. Notes on descriptive set theory and applications to banach spaces th. Descriptive set theory was originally developed on polish spaces. The roots of descriptive set theory go back to the work of borel, baire. Descriptive set theory kurt godel research center universitat wien. In this rst section, we establish several basic facts about trees which we will later utilize through such reductions. The baire category theorem and applications 15 chapter 3. B the formal definition presupposes a and b are sets. The classical part of descriptive set theory has been mainly created and developed by the russian school between 1917 and 1945. In the rst half, we discuss trees, the corresponding representations of closed sets, borel sets, analytic spaces, injectively analytic spaces, and polish spaces, as well as baire category. This text attempts to present a largely balanced approach, which combines many elements of the different traditions of the subject. Lebesgue to the introduction of projective descriptive set theory. Descriptive set theory encyclopedia of mathematics. Its main theme is the interplay of large cardinals, inner models, forcing and descriptive set theory.
In the rst half of the course, we will use techniques from analysis and set theory, as. Volume 100, pages iiixii, 1637 1980 download full volume. More generally, we could consider subsets of any polish space, i. We also draw the connection between the stability theoretic complexity of. Descriptive set theory, american mathematical society in descriptive set theory we try to avoid these pathologies by concentrating on natural while this is a restricted class of sets it includes most of the sets that arise. Newest descriptivesettheory questions mathematics stack. Descriptive set theory has found applications in harmonic analysis, dynamical systems, functional analysis, and various other areas of mathematics. Classical descriptive set theory alexander kechris. The first chapter consists of the study of the descriptive complexity of the set of banachspaces with the.
Effective descriptive set theory is the branch of descriptive set theory dealing with sets of reals having lightface definitions. This subject was started by the french analysts at the turn of the 20th century, most prominently lebesgue, and, initially, was concerned primarily with establishing regularity. C algebras, classification and descriptive set theory description pdf list of participants in this workshop. Descriptive set theory and harmonic analysis 415 this gives kj,i c u c lebesgue measure 0. Classical descriptive set theory graduate texts in. Kechris 2 topological rigidity of automorphism groups w a. In fact, it has long been understood that this is not the case. Oxtoby 90 is a good reference for the basic material concerning measure and category on the real line. About descriptive form of set worksheet descriptive form of set worksheet. A logic of meaning and synonymy, with fritz hamm, pdf file. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. Classical descriptive set theory indian statistical institute.
It is the study of the structure of definable sets and functions in separable completely metrizable spaces. Check that d defined in all the above examples is a metric. Recent projects include the study of foundational and set theoretic questions, and the application of the methodology and results of descriptive set theory, in classical real analysis, harmonic analysis, dynamical systems especially ergodic theory and topological dynamics, model theory, and combinatorics. Pdf results in descriptive set theory on some represented. D by \one can also give a di erent and simpler proof, using a result. In the thesis, the author traces the historical development of descriptive set theory from the work of h. Buy set theory, with an introduction to descriptive set theory studies in logic and the foundations of mathematics vol 86 on free shipping on qualified orders. Descriptive set theory has found applications in harmonic analysis, dynamical systems. Lectures notes for an advanced course given in esslli 2010. Kuratowski 59 and kuratowski and mostowski 60 are excellent references for classical descriptive set theory. In the second half, we establish various relatives of the g. This includes equivalence relation theory, descriptive graph theory, dynamical systems, polish groups, etc. Topics covered include corel lattices, universal sets, the operation a, analytic sets, coanalytic sets, and the continuum hypothesis the appendix.
Notes on descriptive set theory and applications to banach. With an introduction to descriptive set theory sciencedirect. Descriptive set theory, american mathematical society in descriptive set theory we try to avoid these pathologies by concentrating on natural while this is a restricted class of sets. This version, posted on august 5, 2014, includes the material covered in 220ab in. What appeals to me most about descriptive set theory is that to study it you must reallyunderstandso many things. There are two other important examples of such spaces which will play. It is not hard to see that for a perfect set p, every neighborhood of a point p 2p contains in.
Descriptive set theory begins with the study of polish spaces and their borel sets a polish space is a secondcountable topological space that is metrizable with a complete metric. Moschovakis 89 and kechris 54 are more modern treatments of descriptive set theory. An introduction to descriptive set theory indian statistical institute. Descriptive set theory and model theory second lecture. Descriptive form of set worksheet is much useful to the students who woulds like to practice problems on set theory. Past workshops april 7, 2012, at cornell university the sixteenth workshop, which was led by hugh woodin, was on the hod dichotomy. Department of mathematics university of maryland may 6, 2005 in the early days of the development of model theory it was considered natural and was certainly bene. Write the following sets in descriptive form i a a, e, i, o, u ii b 1, 3, 5, 7, 9, 11. Kechris 3 coarse geometry of polish groups 4 geometry of automorphism groups christian rosendal descriptive set theory and model theory notre dame, june 2016 2 22. A polish group is amenable if every continuous action on a compact space admits an invariant borel. Particular texts are more suited for certain directions.