Turing incomparability in Scott sets

Turing Incomparability in Scott Sets

For every Scott set F and every nonrecursive set X in F , there is a Y ∈ F such that X and Y are Turing incomparable.

A measure-theoretic proof of Turing incomparability

We prove that if S is an ω-model of weak weak König’s lemma and A ∈ S, A ⊆ ω, is incomputable, then there exists B ∈ S, B ⊆ ω, such that A and B are Turing incomparable. This extends a recent result of Kučera and Slaman who proved that if S0 is a Scott set (i.e. an ω-model of weak König’s lemma) and A ∈ S0, A ⊆ ω, is incomputable, then there exists B ∈ S0, B ⊆ ω, such that A and B are Turing in...

Lattices of Scott-closed sets

A dcpo P is continuous if and only if the lattice C(P ) of all Scottclosed subsets of P is completely distributive. However, in the case where P is a non-continuous dcpo, little is known about the order structure of C(P ). In this paper, we study the order-theoretic properties of C(P ) for general dcpo’s P . The main results are: (i) every C(P ) is C-continuous; (ii) a complete lattice L is iso...

Representing Scott sets in algebraic settings

Scott's problem for Proper Scott sets

Some 40 years ago, Dana Scott proved that every countable Scott set is the standard system of a model of PA. Two decades later, Knight and Nadel extended his result to Scott sets of size ω1. Here, I show that assuming the Proper Forcing Axiom (PFA), every proper Scott set is the standard system of a model of PA. I define that a Scott set X is proper if it is arithmetically closed and the quotie...