Let κ be an infinite cardinal. A subset of ( κ κ) n is a -subset if it is the projection p[T] of all cofinal branches through a subtree T of (>κ κ) n+1 of height κ. We define and -subsets of ( κ κ) n as usual.
Given an uncountable regular cardinal κ with κ = κ <κ and an arbitrary subset A of κ κ, we show that there is a <κ-closed forcing ℙ that satisfies the κ +-chain condition and forces A to be a -subset of κ κ in every ℙ-generic extension of V. We give some applications of this result and the methods used in its proof.
(i) Given any set x, we produce a partial order with the above properties that forces x to be an element of L .
(ii) We show that there is a partial order with the above properties forcing the existence of a well-ordering of κ κ whose graph is a -subset of κ κ × κ κ.
(iii) We provide a short proof of a result due to Mekler and Väänänen by using the above forcing to add a tree T of cardinality and height κ such that T has no cofinal branches and every tree from the ground model of cardinality and height κ without a cofinal branch quasi-order embeds into T.
(iv) We will show that generic absoluteness for -formulae (i.e., formulae with parameters which define -subsets of κ κ) under <κ-closed forcings that satisfy the κ +-chain condition is inconsistent.
In another direction, we use methods from the proofs of the above results to show that - and -subsets have some useful structural properties in certain ZFC-models.
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