Tukey-order with models on Palikowski's theorems (Set Theory : Reals and Topology)

Abstract

In [Paw86] Pawlikowski proved that, if r is a random real over N, and c is Cohen real over N[r], then (a) in N[r][c] there is a Cohen real over N[c], and (b) 2[ω] ∧ N[c] ∉ N ∧ N[r][c], so in N[r][c] there is no random real over N[c]. To prove this, Pawlikowski proposes the following notion: Given two models N ⊆ M of ZFC, we associate with a cardinal characteristic ξ of the continuum, a sentence ξ[M][N] saying that, in M, the reals in N give an example of a family fulfilling the requirements of the cardinal. So to prove (a) and (b), it suffices to prove that (a') cov(M)[M][[c]][N][[c]] ⇒ cof(M)[M][N] ⇒ cov(N)[M][N'], and (b') cov(M)[M][N] ⇒ add(M)[M][N] ⇒ non(M)[M][[c]][N][[c]] ⇒ cov(N)[M][[c]][N][[c]]. In this paper we introduce the notion of Tukey-order with models, which expands the concept of Tukey-order introduced by Vojtáš [Voj93], to prove expressions of the form ξ[M}[N] ⇒ η[M][N]. In particular, we show (a') and (b') using Tukey-order with models.