Σ11-definability at uncountable regular cardinals

Abstract

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 ( ^{< κ}κ)ⁿ⁺¹ of height κ. We define Σ¹_k-, Π¹_k- and Δ¹_k-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.