# Descriptive set theory by Yiannis N. Moschovakis

By Yiannis N. Moschovakis

Descriptive Set concept is the research of units in separable, entire metric areas that may be outlined (or constructed), and so could be anticipated to have distinctive houses no longer loved by way of arbitrary pointsets. This topic used to be began through the French analysts on the flip of the 20 th century, so much prominently Lebesgue, and, before everything, used to be involved basically with setting up regularity homes of Borel and Lebesgue measurable services, and analytic, coanalytic, and projective units. Its quick improvement got here to a halt within the past due Thirties, basically since it bumped opposed to difficulties that have been self sustaining of classical axiomatic set thought. the sphere turned very energetic back within the Sixties, with the advent of robust set-theoretic hypotheses and strategies from good judgment (especially recursion theory), which revolutionized it. This monograph develops Descriptive Set idea systematically, from its classical roots to the fashionable ``effective'' concept and the results of robust (especially determinacy) hypotheses. The publication emphasizes the principles of the topic, and it units the level for the dramatic effects (established because the Nineteen Eighties) bearing on huge cardinals and determinacy or permitting functions of Descriptive Set thought to classical arithmetic. The ebook contains all of the worthwhile historical past from (advanced) set idea, good judgment and recursion thought

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**Example text**

10) Hint. 18, together with the fact that every closed subset of N × is a countable intersection of clopen sets. 1H. Historical and other remarks 1 The early papers in descriptive set theory were all concerned with sets and functions in real n-space. It was quickly recognized, however, that most results generalized easily at least to Polish spaces, and soon two tendencies developed: one was to stick with the reals or the irrationals and prove the strongest possible results, the other to aim for the widest context in which the basic facts can be established.

Hint. For the ﬁrst assertion, suppose P is the projection of some closed subset C of X ×N . 1, there is a continuous surjection f : N C . Now P is the image of N under f followed by the continuous projection function. We cannot replace N by an arbitrary perfect product space in this result, because of the next exercise. 12 for a related characterization of Σ 11 . 7. Prove that if f : R → X is continuous and F is a closed set of reals, the f[F ] is Σ 02 . Hint. R is a countable union of compact sets.

We will however keep the symbol ∩ for denoting the general set theoretic operation of intersection, with A ∩ B deﬁned for arbitrary sets A, B. Similarly, the disjunction P ∨ Q of two pointsets is deﬁned when P and Q are subsets of the same X and P ∨ Q = P ∪ Q = {x : P(x) ∨ Q(x)}. Negation is most conveniently regarded as a collection of operations ¬X , one for each product space X , with ¬X P deﬁned when P ⊆ X : ¬X P = X \ P = {x ∈ X : ¬P(x)}. In practice we will always write ¬P for ¬X P, as X is clear from the context.