By Arnold W. Miller

A complicated graduate path. a few wisdom of forcing is believed, and a few hassle-free Mathematical good judgment, e.g. the Lowenheim-Skolem Theorem. A pupil with one semester of mathematical good judgment and 1 of set concept might be ready to learn those notes. the 1st part bargains with the overall zone of Borel hierarchies. What are the potential lengths of a Borel hierarchy in a separable metric house? Lebesgue confirmed that during an uncountable whole separable metric area the Borel hierarchy has uncountably many special degrees, yet for incomplete areas the answer's autonomous. the second one part comprises Harrington's Theorem - it's constant to have units at the moment point of the projective hierarchy of arbitrary measurement below the continuum and an evidence and appl- ications of Louveau's Theorem on hyperprojective parameters.

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**Additional resources for Descriptive Set Theory and Forcing: How to prove theorems about Borel sets the hard way**

**Example text**

Similarly to the other argument we see that for any countable E we can choose a countable Q Q X such for any s £T with 2 < r(s) = β < λ (so s ψ ()) we have that [ x G Us \ is not J Hence ord(B) = λ. ) = ω i w e postpone until section 12. 30 9 9 BOREL ORDER OF A FIELD OF SETS Borel order of a field of sets In this section we use the Sikorski-Loomis representation theorem to transfer the abstract Borel hierarchy on a complete boolean algebra into a field of sets. A family F C P(X) is a σ-field iff it contains the empty set and is closed under countable unions and complements in X.

Consequently, it is enough to find 0 sufficiently generic filters for F* Q. 1 0 it is enough to see that if F* QC I is dense in the ccc cBa algebra IB, then B is count ably generated. Let C = {\BCUn]:BeB,neω}. 37 Descriptive Set Theory and Forcing We claim that C generates 1. To see this, note that for each p £ Ψ [χPennun 1= nξω \xp€Un\= B Σ I £ furthermore (p,

Contains the empty set, 2. / is closed under countable unions, 3. AC B e I and A G F implies A el, and 4. X <£ I. e. A « B iff AAB G /. For A G F we use [A] or [A]/ to denote the equivalence class of A modulo /. 1 (Sikorski,Loomis, see [98] section 29) For any countably complete boolean algebra B there exists a σ-field F and a σ-ideal I such that B is isomorphic to F/I. proof: Recall that the Stone space of B, stone(£), is the space of ultrafilters u on B with the topology generated by the clopen sets of the form: [6] = {u G stone(£) :f»G«}.