TY - JOUR

T1 - Robinson forcing is not absolute

AU - Manevitz, Larry Michael

PY - 1976/9

Y1 - 1976/9

N2 - Robinson (or infinite model theoretic) forcing is studied in the context of set theory. The major result is that infinite forcing, genericity, and related notions are not absolute relative to ZFC. This answers a question of G. Sacks and provides a non-trivial example of a non-absolute notion of model theory. This non-absoluteness phenomenon is shown to be intrinsic to the concept of infinite forcing in the sense that any ZFC-definable set theory, relative to which forcing is absolute, has the flavor of asserting self-inconsistency. More precisely: If T is a ZFC-definable set theory such that the existence of a standard model of T is consistent with T, then forcing is not absolute relative to T. For example, if it is consistent that ZFC+ "there is a measureable cardinal" has a standard model then forcing is not absolute relative to ZFC+ "there is a measureable cardinal." Some consequences: 1) The resultants for infinite forcing may not be chosen "effectively" in general. This answers a question of A. Robinson. 2) If ZFC is consistent then it is consistent that the class of constructible division rings is disjoint from the class of generic division rings. 3) If ZFC is consistent then the generics may not be axiomatized by a single sentence of L w/w.

AB - Robinson (or infinite model theoretic) forcing is studied in the context of set theory. The major result is that infinite forcing, genericity, and related notions are not absolute relative to ZFC. This answers a question of G. Sacks and provides a non-trivial example of a non-absolute notion of model theory. This non-absoluteness phenomenon is shown to be intrinsic to the concept of infinite forcing in the sense that any ZFC-definable set theory, relative to which forcing is absolute, has the flavor of asserting self-inconsistency. More precisely: If T is a ZFC-definable set theory such that the existence of a standard model of T is consistent with T, then forcing is not absolute relative to T. For example, if it is consistent that ZFC+ "there is a measureable cardinal" has a standard model then forcing is not absolute relative to ZFC+ "there is a measureable cardinal." Some consequences: 1) The resultants for infinite forcing may not be chosen "effectively" in general. This answers a question of A. Robinson. 2) If ZFC is consistent then it is consistent that the class of constructible division rings is disjoint from the class of generic division rings. 3) If ZFC is consistent then the generics may not be axiomatized by a single sentence of L w/w.

UR - http://www.scopus.com/inward/record.url?scp=51649186188&partnerID=8YFLogxK

U2 - 10.1007/BF02757001

DO - 10.1007/BF02757001

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AN - SCOPUS:51649186188

SN - 0021-2172

VL - 25

SP - 211

EP - 232

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

IS - 3-4

ER -