### Σ11-definability at uncountable regular cardinals

Philipp Lücke
Source: J. Symbolic Logic Volume 77, Issue 3 (2012), 1011-1046.

#### 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 ( < 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.

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Permanent link to this document: http://projecteuclid.org/euclid.jsl/1344862172
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Mathematical Reviews number (MathSciNet): MR2987148

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