By Kanovei V., Sabok M., Zapletal J.

This publication lays the principles for a thrilling new region of analysis in descriptive set idea. It develops a powerful connection among energetic issues: forcing and analytic equivalence kin. This in flip permits the authors to enhance a generalization of classical Ramsey thought. Given an analytic equivalence relation on a Polish house, can one discover a huge subset of the distance on which it has an easy shape? The e-book offers many optimistic and unfavorable normal solutions to this question. The proofs function right forcing and Gandy-Harrington forcing, in addition to partition arguments. the consequences contain robust canonization theorems for lots of periods of equivalence relatives and sigma-ideals, in addition to ergodicity leads to circumstances the place canonization theorems are very unlikely to accomplish. excellent for graduate scholars and researchers in set thought, the ebook presents an invaluable springboard for extra study

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Even though at some point in its cumulative hierarchy the model N may not be wellfounded, we will always identify the wellfounded initial part of N with its transitive isomorph. 48 If I is a σ -ideal then j Hℵ1 = id. Also, in such a case j (ω) = ω, j (2ω ) = 2ω ∩ N and j (ωω ) = ωω ∩ N . Proof We will first argue that ω N = j ω. To verify the equality, return to the ground model and assume that S ∈ R I is a condition forcing [ f ] ∈ ω N . We may thin the set S if necessary so that S = dom( f ) and f S ⊂ ω.

It may happen that there is an analytic set A such that j (A) = Aφ ∩ j (X ). For example, if there is an illfounded countable ordinal β ∈ N and A is the set of illfounded linear orders on ω, then N contains some linear ordering x on ω which is isomorphic to β, and / j (A) (as N |= x is isomorphic to then x ∈ Aφ (as β is illfounded) and x ∈ an ordinal). In such a case, the model N may be incorrect about such items as codes for Borel sets, etc. Proof The argument is entirely standard. Let I be the σ -ideal on nonstationary subsets of [Hλ ]ℵ0 , and let R I be the quotient poset.

Since these points are generic for the product PI × PI over the model M, there must be a condition Bx , B y ∈ (PI × PI ) ∩ M which forces the ˙ and moreover x ∈ Bx , y ∈ B y and Bx ∩ B y = 0. generic pair to belong to D, Since the intersection B y ∩ C is nonempty, it is in fact I -positive. Whenever z ∈ B y ∩ C is a point, then the PI × PI -generic filter over M determined by the pair x, z contains the condition Bx , B y . Therefore by the forcing theorem M[x, z] |= x, z ∈ D and by analytic absolutenes between the model M[x, y] and V , x, z ∈ D.