Factoring Solovay-random extensions, with application to the reduction property
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Factoring Solovay-random extensions, with application to the reduction property Vladimir Kanovei1
· Vassily Lyubetsky1
Received: 30 November 2018 / Accepted: 3 November 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract If a real a is random over a model M and x ∈ M[a] is another real then either (1) x ∈ M , or (2) M[x] = M[a], or (3) M[x] is a random extension of M and M[a] is a random extension of M[x]. This result may belong to the old set theoretic folklore. It appeared as Exapmle 1.17 in Jech’s book “Multiple forcing” without the claim that M[x] is a random extension of M in (3), but, likely, it has never been published with a detailed proof. A corollary: 1n -Reduction holds for all n ≥ 3, in models extending the constructible universe L by κ -many random reals, κ being any uncountable cardinal in L. Keywords Forcing · Solovay-random extensions · Factoring · Reduction property Mathematics Subject Classification 03E35 · 03E15
1 Introduction It is known from Solovay [20], and especially Grigorieff [3] in most general form, that any subextension V[x] of a generic extension V[G], generated by a set x ∈ V[G], is itself a generic extension V[x] = V[G 0 ] of the same ground universe V, and the whole extension V[G] is equal to a generic extension V[G 0 ][G 1 ] of the intermediate model V[x] = V[G 0 ]. See a more recent treatment of this question in [5,9,13,21].
Communicated by S.-D. Friedman. This research was funded by Russian Foundation for Basic Research RFBR Grant Number 18-29-13037.
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Vladimir Kanovei [email protected] Vassily Lyubetsky [email protected]
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Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow, Russia
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V. Kanovei, V. Lyubetsky
In particular, it is demonstrated in [9] that if P = P ; ≤ ∈ V is a forcing notion, a set G ⊆ P is P-generic over V, t ∈ V is a P-name, x = t[G] ∈ V[G] is the G -valuation of t , and x ⊆ V, then (1) there is a set ⊆ P such that V[] = V[x] and G is -generic over V[x]; (2) there exists an order ≤t on P in the ground universe V, such that p ≤ q implies p ≤t q , and itself is P ; ≤t -generic over V. However the nature and forcing properties of the derived forcing notions, that is, P0 = P ; ≤t ∈ V and P1 (x) = ; ≤ ∈ V[x], is not immediately clear. At the trivial side, we have the Cohen forcing P = C = 2 0. It holds in V that there is an Fσ set A 2 ⊆ A2 of the same measure μ0 (A 2 ) = μ0 (A2 ). The Borel set A2 A 2 , coded in V, is null, and hence a0 ∈ A 2 . Therefore there is, in V, a perfect set A3 ⊆ A 2 , satisfying a0 ∈ A3 and μ0 (A3 ) > 0. The set R of all open rational intervals J ⊆ I such that μ0 (A3 ∩ f 0 −1 [J ]) = 0 is at most countable. Therefore A0 = A3 J ∈R f 0 −1 [J ] is a closed subset of A3 , of the same measure μ0 (A0 ) = μ0 (A3 ) > 0 — hence a0 ∈ A0 (by the randomness). To simplify things, define the restricted function f = f 0 A0 . Then f maps A0 onto the closed set Y0 = f 0 [A0 ] = f [A0 ] (since generally continuous images of compact sets a
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