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Descriptive Set Theory Descriptive Set theory by George Barmpalias Institute for Language, Logic and Computation Lectures 1–8 Last updated on the 10th of November 2009 Answer I The study of sets of real numbers that can be described. I . sets of reals that can be defined I Definability theory What is Descriptive set theory? What is Descriptive set theory? Answer I The study of sets of real numbers that can be described. I . sets of reals that can be defined I Definability theory Answer I Analysis, topology I Set theory I Computability / recursion theory Although the continuum is our primary concern, a lot of the results hold for more general spaces. Also the study of the continuum often requires the study of more general spaces. This is why descriptive set theory studies abstract spaces that are ‘similar’ to the continuum. Polish spaces: complete, separable metric spaces with no isolated points (perfect). What (meta) mathematics are involved in it? What (meta) mathematics are involved in it? Answer I Analysis, topology I Set theory I Computability / recursion theory Although the continuum is our primary concern, a lot of the results hold for more general spaces. Also the study of the continuum often requires the study of more general spaces. This is why descriptive set theory studies abstract spaces that are ‘similar’ to the continuum. Polish spaces: complete, separable metric spaces with no isolated points (perfect). In early 1900s analysts Lebesgue, Suslin, Lusin, Borel (and others) applied his ideas to analysis. the classification and description of pointsets in classical analysis It was then that Cantor’s ideas received general acceptance Birth of Descriptive set theory Cantor discovered higher infinities (late 1800s) in an attempt to solve a problem on zeros of a Fourier series. This was not immediately appreciated. Birth of Descriptive set theory Cantor discovered higher infinities (late 1800s) in an attempt to solve a problem on zeros of a Fourier series. This was not immediately appreciated. In early 1900s analysts Lebesgue, Suslin, Lusin, Borel (and others) applied his ideas to analysis. the classification and description of pointsets in classical analysis It was then that Cantor’s ideas received general acceptance Fathers of Descriptive set theory (early 1900s) Timeline: complexity of the continuum Pythagoreans discovered the irrational numbers. Only in the 19th century the continuum was conceived as a complete ordered field. The axiom of completeness supplied the concept of limits, intermediate value theorem, mean value theorem etc. essentially the ingredients for the success of the calculus in the 19th century. This axiom also opened a whole Pandora’s box of set- theoretical difficulties in the 20th century. Cantor’s program Cantor started studying perfect sets (closed sets without isolated points) in connection to a problem in Fourier series Later he studied them in order to settle the continuum hypothesis. He hoped to show that every uncountable set contains a perfect set. Cantor’s program It was known that every perfect set has cardinality 2@0 . Also, every uncountable Borel set contains a perfect subset. Lusin (1921) showed that under the axiom of choice, Cantor’s program fails. Non-uniqness of the real line In the 1960s Cohen’s method of forcing showed a number of independence results illustrating the non-uniqness of the real line. There are natural question about the reals whose answer depends on additional axioms in set theory. These questions arise because the completeness axiom of the continuum forces us to introduce non-absolute concepts into analysis. like the concept of power sets. It became a task of descriptive set theory to discover just where in the various hierarchies of complexity the disease occurs. Effective theory Independently of the classical developments, logician Stephen Kleene studied the definability of sets of natural numbers in the 1940s and 1950s This theory presented many similarities with the classical theory of definability of sets of reals. Classical Effective Reals integers continuous functions computable functions Borel sets Hyperarithmetical sets 1 analytic sets Σ1 sets projective sets analytical sets In this course we will study the two theories in parallel. Among the topics will be: I Definability hierarchies (Borel, Baire, Projective etc.) I Separation theorems I Games and determinacy I Hyperarithmetic theory I Equivalence relations . with lots of examples, history and exercises. I Assessment: 80% assignments + 20% exam I Methodology: mathematical I Course webpage: http://www.barmpalias.net/Descr.html Bibliography I Recursive aspects of descriptive set theory, by R. Mansfield and G. Weitkamp (Oxford logic guides) I Descriptive set theory, by Y. Moschovakis (Studies in logic and the FOM) I Classical descriptive set theory, by A. Kechris (Graduate texts in Mathematics) I Descriptive set theory notes, by David Marker: http://www.math.uic.edu/ marker/math512/dst.pdf The Baire space N is the set of all sequences of natural numbers. It has a natural topology generated by the basic open sets Nσ = fα j σ ⊂ αg. These are clopen (both closed and open) sets in N . There is a natural metric for this space: d(α; β) = 1=the least position where they differ and d(α; β) = 0 if α = β. The Baire space is separable i.e. it has a countable dense subset. Show that the Baire space is not compact. The Cantor space is compact (see below). The Baire space N is homeomorphic to the irrationals via continued fractions. A homeomorphism is a continuous function between two topological spaces with a continuous inverse function. A tree in N is a set of finite sequences which is closed downward under initial segments. Note: Every closed set in the Baire or Cantor space can be represented as the set of infinite paths through a tree. Theorem (König’s Lemma) In the Baire space, a finite branching tree has an infinite path iff it is infinite. Proof. I If the tree is finite, obviously it does not have infinite paths. I If it is infinite, we construct an infinite path by induction: I Starting with the empty sequence, inductively assume that β n is defined and has the property that there are infinitely many strings on the tree which extend it. I Since the tree is finite branching, there is m 2 N such that there are infinitely many strings on the tree which extend (β n) ∗ m. I Pick such n and let β n + 1 = (β n) ∗ m. I By induction, β is an infinite path in the tree. Corollary The Cantor space C is compact. Proof. I Let fσi g be an open cover of the Cantor space (assume 8i σi 6= ;). I Consider the set of binary strings τ such that 8i; σi 6⊆ τ. I This set of strings is a tree T . without infinite paths I Therefore it is finite and it has finitely many ‘leafs’ (maximal nodes). I the immediate successors of those ‘leafs’ are also finitely many and belong to the original cover of C. I The set of their immediate successors form the required finite sub-cover of the space. I It covers C: any point in its complement would have all of its initial segments prefixed by some σi . Corollary The clopen sets in the Cantor space are exactly the finite unions of the basic open sets Nσ. Proof. I Each Nσ is both open and closed (easy to show). I Therefore their finite unions are both open and closed. I For the other direction, let A be a clopen set. I Then A = [i Nσi and C − A = [i Nτi for suitable families of strings. I Hence C = ([i Nσi ) [ ([i Nτi ) and the union of these families is a cover of C. I By compactness C = ([i<k Nσi ) [ ([i<k Nτi ) for some k 2 ! I Obviously [i<k Nσi ⊆ A. I But A ⊆ [i<k Nσi because if α 2 A then α 62 [i<k Nτi , so α 2 [i<k Nσi . Task Show that there is a clopen set in the Baire space which is not a finite union of basic open sets. The language we use to talk about N and ! has I Variables for integers I Variables for reals (Greek letters) I constants for natural numbers I +, ·, exp I function applications e.g. α(x + 3) I equalities, inequalities I quantifiers (first order, second order), negations, conjunctions, disjunctions Examples 1 8α9n [α(n + m + 1) < 5] This is a Π1 formula. It has a second order universal quantifier in front of a first order existential quantifier. Two universal quantifiers amount to one, through standard coding. Same for existential.. So we only count quantifier alternations. 1 9β8α8n9m (m > n ^ β(α(n)) = Y (m + 1)) This is a Σ2 formula. n 0 8n (n < 2 ) is a Π1 formula. 1 1 The negation of a Π1 formula is Σ1; and vice-versa. 1 1 A set A is Σ2 if it can be defined by a Σ2 formula. 1 That is, A = fX j P(X)g where P is a Σ2 formula with free variable X. 1 1 1 The sets that are both in Σm and Πm are called ∆m. 0 0 0 Similarly, the sets that are both in Σm and Πm are called ∆m. 0 0 0 The arithmetical hierarchy consists of the classes Σm, Πm, ∆m. 1 1 1 The analytical hierarchy consists of the classes Σm, Πm, ∆m. Formal definition 0 A formula in arithmetic is Σn if it can be written as a sentence with no quantifiers, prefixed by an alternating string of n first-order quantifiers starting with 9. A formula is arithmetical if it is in the arithmetical hierarchy, i.e.
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