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Introduction to Classical Descriptive Set Theory Early Development Definable Sets of Reals Regularity Properties Metamathematical Issues References Introduction to Classical Descriptive Set Theory J.T. Chipman Logic Seminar Stanford University 13 October 2015 J.T. Chipman Logic Seminar Stanford University Introduction to Classical Descriptive Set Theory Early Development Definable Sets of Reals Regularity Properties Metamathematical Issues References \The efforts that I exerted led me to the following totally unexpected discovery: there exists a family [...] consisting of effective [i.e., definable] sets, such that one does not know and one will never know whether every set from this family, if uncountable, has the cardinality of the continuum, nor whether it [has the property of Baire], nor whether it is measurable. [...] This is the family of the projective sets of Mr. H. Lebesgue. It remains but to recognize the nature of this new development." (Luzin (1925)) J.T. Chipman Logic Seminar Stanford University Introduction to Classical Descriptive Set Theory Early Development Definable Sets of Reals Regularity Properties Metamathematical Issues References Table of contents 1 Early Development 2 Definable Sets of Reals Borel Sets Projective Sets L(R) 3 Regularity Properties Lebesgue Measurability Property of Baire Perfect Set Property Distribution of Regularity Properties 4 Metamathematical Issues 5 References J.T. Chipman Logic Seminar Stanford University Introduction to Classical Descriptive Set Theory Early Development Definable Sets of Reals Regularity Properties Metamathematical Issues References Kanamori summarizes the subject matter of Descriptive Set Theory succinctly: \Descriptive Set Theory is the definability theory of the continuum." Concerning the origins of the theory, Moschovakis writes in the opening lines of his textbook: \The roots of Descriptive Set Theory go back to the work of Borel, Baire, and Lebesgue around the turn of the 20th century, when the young French analysts were trying to come to grips with the abstract notion of a function. " J.T. Chipman Logic Seminar Stanford University Introduction to Classical Descriptive Set Theory Early Development Definable Sets of Reals Regularity Properties Metamathematical Issues References In this section, we provide a brisk history of the relevant results of Borel, Baire, Lebesgue, and, last, the Russians Luzin and Suslin. There are three goals of this presentation: To provide a brief historical and technical introduction to the study of definable sets of reals and their regularity properties To make explicit the relevant mathematical motivations of this theory (as opposed to logical or metamathematical motivations), and To motivate the relationship between definability and principles of definable determinacy J.T. Chipman Logic Seminar Stanford University Introduction to Classical Descriptive Set Theory Early Development Definable Sets of Reals Regularity Properties Metamathematical Issues References Emile´ Borel (1871-1956) Borel's 1898 book, Le¸conssur la th´eorie des fonctions (\Lessons on the theory of functions"), introduced the contemporary notion of a measure on the real line: The development of analytical geometry motivated the need n to define a general notion of measure for (subsets of) R One of Borel's key new ideas was employing countable additivity and using this to define the Borel sets Beginning with a family of sets of an obvious measure, such as open or closed intervals on [0; 1], and a function assigning to each interval its length, he recursively expands the function's domain of definition in stages to sets whose complements are previously defined and sets that are the unions of previously defined disjoint sets J.T. Chipman Logic Seminar Stanford University Introduction to Classical Descriptive Set Theory Early Development Definable Sets of Reals Regularity Properties Metamathematical Issues References Ren´e-LouisBaire (1874-1932) Baire's 1899 thesis, Sur le fonctions de variables r´eelles (\On the functions of real variables"), analyzed real functions in terms of a proper transfinite hierarchy|the first such analysis since Cantor: Baire class 0 consists of the continuous real functions For countable ordinals α > 0, Baire class α consists of functions f not in any previous class but are nonetheless pointwise limits of sequences of functions g0; g1; g2; ::: from previous classes: i.e., f (x) = limn!1gn(x). Further, he proved two facts concerning the functions in these classes (i.e., the Baire functions): They are closed under pointwise limits, and There exist non-Baire real functions J.T. Chipman Logic Seminar Stanford University Introduction to Classical Descriptive Set Theory Early Development Definable Sets of Reals Regularity Properties Metamathematical Issues References Henri Lebesgue (1875-1941) Lebesgue's 1902 thesis, Int´egral,longueur, aire (\Integral, length, area"), is the source of modern measure theory. His concept of measurable set subsumed the Borel sets and his analytic definition of measurable function subsumed the Baire functions. Further elaboration of these ideas in his 1905 Sur les fonctions repr´esentablesanalytiquement (\On the analytically representable functions") yielded two major results (stated on the following page) of enduring interest. These results were enabled by establishing a correlation between the Borel sets and the Baire functions: the former are exactly the pre-images fxjf (x) 2 Og of open intervals O by Baire functions f . J.T. Chipman Logic Seminar Stanford University Introduction to Classical Descriptive Set Theory Early Development Definable Sets of Reals Regularity Properties Metamathematical Issues References Here are the theorems: For every countable ordinal α, there is a Baire function of class α. I.e., the Baire (and hence Borel) hierarchy is proper. There is a Lebesgue measurable function that is not in any Baire class. Hence there are non-Borel Lebesgue measurable sets. Between 1905 and 1915, the work of the French analysts also resulted in a codification of the regularity properties|properties indicative of \well-behaved" sets of reals. It would be, however, a Russian student in Paris, Nikolai Luzin, whose initiative led to the subject of descriptive set theory as a distinct discipline. J.T. Chipman Logic Seminar Stanford University Introduction to Classical Descriptive Set Theory Early Development Definable Sets of Reals Regularity Properties Metamathematical Issues References Nikolai Luzin (1883-1950) & Mikhail Suslin (1894-1919) While investigating the results of Lebesgue described above, one of Luzin's students, Suslin, made an important discovery concerning k+1 the projections of Borel sets. I.e., for Y ⊆ R , the projection of Y is: k p[Y ] = f< x1; :::; xk >2 R j9y(< x1; :::; xk ; y >2 Y )g Suslin discovered a mistaken assumption of Lebesgue's to the effect that projections of (Gδ) Borel subsets of the plane are also Borel by constructing a counterexample. Hence the Borel sets are not closed under projection (it turns out that, more generally, they are not closed under continuous images). J.T. Chipman Logic Seminar Stanford University Introduction to Classical Descriptive Set Theory Early Development Definable Sets of Reals Regularity Properties Metamathematical Issues References Suslin's 1917 Sur une d´efinition des ensembles mesurables B[orel] sans nombres transfinis (\On a definition of Borel measurable sets without transfinite numbers") uses something he calls \operation A" to directly define the sets used in his counterexample|the analytic sets. We will, however, use the more modern definition of them: ! X ⊆ R is analytic if there is a closed Y ⊆ R × ! such that, for all x, x 2 X $ 9y 2 !!hx; yi 2 Y . In effect, operation A had allowed Suslin to \factor out" and collect the first coordinates of the ordered pairs in countable intersections of open sets in two dimensions by means of uncountable unions. J.T. Chipman Logic Seminar Stanford University Introduction to Classical Descriptive Set Theory Early Development Definable Sets of Reals Regularity Properties Metamathematical Issues References Suslin's work on analytic sets produced four important theorems: Every Borel set is analytic There is an analytic set that is not Borel A set of reals is Borel iff both it and its complement are analytic Every analytic set is Lebesgue measurable, has the property of Baire, and has the perfect set property Note the analogy here, for the first three, to later results. I.e., when \Borel" is replaced by \recursive" and \analytic" is replaced by \recursively enumerable." We will return to this point later. J.T. Chipman Logic Seminar Stanford University Introduction to Classical Descriptive Set Theory Early Development Definable Sets of Reals Regularity Properties Metamathematical Issues References In the rest of the presentation, we use the preceding historical survey to motivate central concerns of contemporary descriptive set theory: The hierarchy of definable sets of reals: Borel, Projective, and those in L(R) Regularity properties of definable sets of reals: Lebesgue measurability, Baire property, and the perfect set property Note that I omit the \structural properties," such as Uniformization, for the sake of time The limitative results in the field and motivating principles of definable determinacy J.T. Chipman Logic Seminar Stanford University Introduction to Classical Descriptive Set Theory Early Development Definable Sets of Reals Regularity Properties Metamathematical Issues References 1 Early Development 2 Definable Sets of Reals Borel Sets Projective Sets L(R) 3 Regularity Properties
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