The Complexity of the Dependence Operator

We show that Leitgeb’s dependence operator of [2] is a Π11-operator and that this is best possible. In [2] Hannes Leitgeb introduced a dependence operatorD as a relation between sets of sentences in a language LTr such as that for arithmetic (which we shall take here) augmented with a predicate symbol Ṫr to represent truth. We shall assume the reader has a familiarity with this article and its ideas, as our purpose here is only to correct an error of our own: we had mistakenly persuaded its author that this operator was hyperarithmetic (that is the relation ‘ pσq ∈ D(Φ)’ as a relation of σ and Φ was ∆11). We had realised that this was false, but unfortunately too late for its suppression in [2]. Since a number of people have subsequently asked about this, we feel we should set the record straight, (and take this opportunity to apologise to Hannes Leitgeb for being so misleading). We thus show here: Proposition 1 The relation ‘ pσq ∈ D(Φ)’ is in general Π11 and this is best possible. Indeed if Φα is a stage in the increasing monotone hierarchy of dependency sets building up to the least fixed point Φlf of [2] Section 3, then for any α > 0, Φα is a Π11-complete set of gödel numbers. Proof: The first sentence will follow from the second, so we prove the latter. Recall that we define Φ0 = Ø and then: Φα+1 = { pφq | φ ∈ D(Φα)} = {pφq | φ depends on Φα} = = {pφq | ∀Ψ((N,Ψ) |= φ ↔ (N,Ψ ∩Φα) |= φ)} ; Φλ = ⋃