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

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

Cardinal invariants of the continuum -A survey

Abstract These are expanded notes of a series of two lectures given at the meeting on axiomatic set theory at Kyōto University in November 2000. The lectures were intended to survey the state of the art of the theory of cardinal invariants of the continuum, and focused on the interplay between iterated forcing theory and cardinal invariants, as well as on important open problems. To round off the present written account of this survey, we also include sections on ZF C-inequalities between cardinal invariants, and on applications outside of set theory. However, due to the sheer size of the area, proofs had to be mostly left out. While being more comprehensive than the original talks, the personal flavor of the latter is preserved in the notes. Some of the material included was presented in talks at other conferences

Goldstern–Judah–Shelah preservation theorem for countable support iterations

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