Descriptive set theory by Yiannis N. Moschovakis

By Yiannis N. Moschovakis

Descriptive Set idea is the learn of units in separable, whole metric areas that may be outlined (or constructed), and so might be anticipated to have certain homes no longer loved via arbitrary pointsets. This topic used to be begun by way of the French analysts on the flip of the twentieth century, so much prominently Lebesgue, and, in the beginning, was once involved essentially with developing regularity homes of Borel and Lebesgue measurable features, and analytic, coanalytic, and projective units. Its swift improvement got here to a halt within the overdue Thirties, essentially since it bumped opposed to difficulties which have been self sustaining of classical axiomatic set concept. the sphere turned very lively back within the Nineteen Sixties, with the advent of robust set-theoretic hypotheses and strategies from common sense (especially recursion theory), which revolutionized it. This monograph develops Descriptive Set idea systematically, from its classical roots to the trendy ``effective'' concept and the implications of sturdy (especially determinacy) hypotheses. The e-book emphasizes the rules of the topic, and it units the degree for the dramatic effects (established because the Nineteen Eighties) bearing on huge cardinals and determinacy or permitting functions of Descriptive Set concept to classical arithmetic. The publication comprises all of the valuable history from (advanced) set conception, good judgment and recursion idea

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Hint. The result is known for finite , so we proceed by induction. Choose 0 , 1 , . . so that the supremum{ i + 1 : i = 0, 1, . . } = and let Gi ⊆ N × X be universal for Σ 0i X . Put G(α, x) ⇐⇒ i ¬Gi (α)i , x and show that G is universal for Σ 0 X by verifying that each P in Σ 0 P(x) ⇐⇒ i X satisfies ¬Pi (x) 0 with each Pi in Σ i . In the exercises of the next section we will extend this result to show that each Σ 0 is Y-parametrized for each perfect product space Y. 7. Suppose R0 , R1 , · · · are all subsets of the same space X and we take R(x, s) ⇐⇒ Rs (x).

Now the set G ⊆ X × N defined by G (x, α) ⇐⇒ G x, α(0) is also open since G is and the map (x, α) → x, α(0) is continuous, hence F is Π 11 . Thus every closed set is Π 11 and then, by definition, every Σ 11 set is Σ 12 , from which Σ 1n ⊆ Σ 1n+1 follows immediately by induction. The remaining inclusions in the diagram are trivial. 2] 1E. The projective sets 31 To prove the closure properties of the Lusin pointclasses we need maps that allow us to code infinite sequences of irrational by single irrationals.

3. Let f : R → R be a continuous function on the line. Prove that the relations P(x, y) ⇐⇒ f (x) = y, Q(x) ⇐⇒ f (x) exists are both Π 03 . Hint. Let r0 , r1 , . . be an enumeration of all rational numbers and put R(x, y, s, k, m) ⇐⇒ rm = 0 & 1 1 . 6] 1C. 5. Uniformization. Clearly R is a closed relation. It is easy to verify that 1 =⇒ R(x, y, s, k, m) . k+1 The second assertion is proved similarly, starting with the relation P(x, y) ⇐⇒ (∀s)(∃k)(∀m) 0 < |rm | < S(x, s, k, m, n) ⇐⇒ rm = 0 & rn = 0 & 1 rm {f(x + rm ) − f(x)} − 1 rn {f(x + rn ) − f(x)} ≤ 1 s+1 .

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