Let κ be an infinite cardinal. A subset of (κκ)n is a Σ11-subset if it is the projection p[T] of all cofinal branches through a subtree T of ( < κκ)n+1 of height κ. We define Σ1k-, Π1k- and Δ1k-subsets of (κκ)n as usual.
Given an uncountable regular cardinal κ with κ=κ< κ and an arbitrary subset A of κκ, we show that there is a < κ-closed forcing ℛ that satisfies the κ+-chain condition and forces A to be a Δ11-subset of κκ in every ℛ-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(𝒫(κ)).
ii) We show that there is a partial order with the above properties forcing the existence of a well-ordering of κκ whose graph is a Δ12-subset of κκ×κκ.
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 Σ13(κκ)-formulae (i.e., formulae with parameters which define Σ13-subsets of κκ) 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 Σ11- and Δ11-subsets have some useful structural properties in certain ZFC-models.
"Σ11-definability at uncountable regular cardinals." J. Symbolic Logic 77 (3) 1011 - 1046, September 2012. https://doi.org/10.2178/jsl/1344862172