Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

136 concept calculus: much better thanContinue in this way as long as possible. This results in a partial r tuple u anda nondegenerate subinterval I, such that the extensions of u of p, r type σ aredense in I.Let J be a nondegenerate subinterval of I and t be an integer. Suppose J doesnot contain an extension of u, ofp, r type σ , where the new coordinates are all> t. Then the union over the remaining coordinate positions i ≤ r of the extensionsof u of p, r type σ , whose i-th coordinate is ≤ t, is dense in J. Hence, one of thesesets is dense in some nondegenerate subinterval of J. Therefore, we have t ′ ≤ tand a remaining coordinate position i ≤ r such that the extensions of u of p, r typeσ , whose i-th coordinate is t ′ , are dense in some nondegenerate subinterval of J.This contradicts that we could not continue the process. Hence, J does containan extension of u, ofp, r type σ , where the new coordinates are all > t.Lemma 3.4 There is no model (E, S, ∼)ofT 2 , definable in (Z,