A very weak generalization of SPFA to higher cardinals

Itay Neeman found in [7] a new way of iterating proper forcing notions and extended it in [8] to א2. In [5] his construction for א1 ([7]) was generalized to semi-proper forcing notions. We apply here finite structures with pistes in order extend the construction to higher cardinals. In the final model a very weak form of SPFA will hold. 1 Basic definitions and main results The following two definitions are due to S. Shelah [9]. Definition 1.1 A forcing notion Q is called a {η}–proper iff for every M ≺ 〈H(χ),∈, < 〉 of a size η with Q ∈M the following holds: for every q ∈M there is p ≥ q which is (M,Q)-generic, i.e. p ((M [G ∼]) V = M). If Q is {η′}–proper for every regular cardinal η′ ≤ η, then we call Q a {≤ η}–proper. Definition 1.2 A forcing notion Q is called a {η}–semi-proper iff for every M ≺ 〈H(χ),∈, < 〉 of a size η with Q ∈M the following holds: for every q ∈M there is p ≥ q which is (M,Q)–semi-generic, i.e. p (M [G ∼]∩η + = M ∩η). If Q is {η′}–proper for every regular cardinal η′ ≤ η, then we call Q a {≤ η}–semi-proper. Remark 1.3 Further we will use a bit weaker notions. Instead of arbitrary M ’s in Definitions 1.1,1.2 we restrict ourself to models closed under < η–sequences in GCH situations and once GCH breaks down to models which are generic extensions of closed under < η– sequences models from the ground model which satisfies GCH. It is possible to formulate this in terms of internal clubs as Neeman does. ∗Partially supported by ISF grant no. 58/14.