caa a22 4500 388039698 CHVBK 20180307125017.0 cr unu---uuuuu 161130e199803 xx s 000 0 eng 10.2307/2586594 doi S0022481200015395 pii (NATIONALLICENCE)cambridge-10.2307/2586594 Friedman Sy D. Mathematics Department, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA, E-mail: sdf@math.mit.edu Generic saturation [Elektronische Daten] [Sy D. Friedman] Assuming that ORD is ω + ω-Erdös we show that if a class forcing amenable to L (an L-forcing) has a generic then it has one definable in a set-generic extension of L[O #]. In fact we may choose such a generic to be periodic in the sense that it preserve the indiscernibility of a final segment of a periodic subclass of the Silver indiscernibles, and therefore to be almost codable in the sense that it is definable from a real which is generic for an L-forcing (and which belongs to a set-generic extension of L[0 #]). This result is best possible in the sense that for any countable ordinal α there is an L-forcing which has generics but none periodic of period ≤ α. However, we do not know if an assumption beyond ZFC+"O # exists” is actually necessary for these results. Let P denote a class forcing definable over an amenable ground model 〈L, A〉 and assume that O # exists. Definition. P is relevant if P has a generic definable in L[0 #]. P is almost relevant if P has a generic definable in a set-generic extension of L[0 #]. Remark. The reverse Easton product of Cohen forcings 2<κ, κ regular is relevant. So are the Easton product and the full product, provided κ is restricted to the successor cardinals. See Chapter 3, Section Two of Friedman [3]. Of course any set-forcing (in L) is almost relevant. Definition. κ is α-Erdös if whenever C is CUB in κ and f: [C]<ω → κ is regressive (i.e., f(a) < min(a)) then f has a homogeneous set of ordertype α. Copyright © Association for Symbolic Logic 1998 The Journal of Symbolic Logic Cambridge University Press 63/1(1998-03), 158-162 0022-4812 63:1<158 1998 63 JSL https://doi.org/10.2307/2586594 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.2307/2586594 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Friedman Sy D. Mathematics Department, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA, E-mail: sdf@math.mit.edu NATIONALLICENCE 773 0- The Journal of Symbolic Logic Cambridge University Press 63/1(1998-03), 158-162 0022-4812 63:1<158 1998 63 JSL CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-cambridge