T-SEPERATING SETS FOR COHERENT SEQUENCES

In a previous paper the author used methods of Witzany to give a lower bound for the smallest repeat point of a coherent sequence. Here the notion of a T-seperating set is introduced, and the lower bound is improved. AMS Subject Classification: 03E55


Introduction
In [4] some methods are introduced for constructing seperating stationary sets for coherent sequences of normal ultrafilters.Some further results are given in [3].Here these methods are improved on.Superschemes were introduced in [1], and an improved discussion is given in [2].The methods here permit the use of superschemes in constructing seperating stationary sets.
As noted in [3], by results of Mitchell there is a model Notation for coherent sequences will be as in [3].Hereafter in this section it will be assumed that GCH holds and U is maximal, so that for a measurable cardinal κ, Dom(U (κ)) = o(κ) ≤ κ ++ .
It may be easier to construct T-seperating sets than representing functions on [0, Dom(U (κ))), and studying them in their own right is of interest.
where I is the thin ideal.
A scheme is a recipe for an iteration.Given a scheme Σ with σ < Dom(κ) the subset S Σ α may be defined inductively for α ≤ σ as follows.0.

Superschemes
Recall from [2] that for κ ∈ Card a superscheme is a pair Σ = σ, φ where σ < κ ++ and φ is as for a scheme.To use a superscheme for an iteration a method must be specified for obtaining S α when cf(α) = κ + .Some discussion of this problem will be given here.
Suppose α = κ + , U is a normal ultrafilter on κ, and M = Ult U (V ).Since the well-orders of κ, coded as subsets of κ, are the same in V and M , and the order type function is ∆ 1 , (κ + ) M = κ + .Thus, the function λ → λ + represents κ + mod any normal ultrafilter on κ, and a T-seperating set may be constructed using theorem 1.
Another construction may be given by adapting a method of proposition 3.9 if [4].As in [3] let C denote the map α → j U (κ)(α) (κ).C U (κ) may be written to indicate U and κ.

Proof. As in the proof of theorem 2 let
Then g ∈ M , and one verifies (see [4]) that Let C (1) denote the fixed point enumerator of C. If there is an α such that Proof.The proof of theorem 8 may be modified.Let S 1α be a T-seperating set at α. Let f E α be an E-representing function for α.Let S 1σ be a T-seperating set at σ. Let f E σ be an E-representing function for σ.Let S 2 be the set S, as in the proof of theorem 8, with f E α used in place of f α and f E σ used in place of f σ .Then S 1α ∩ S 1σ ∩ S 2 is T-seperating at C (1) (α).
These methods can be pursued further.Whether there are methods which may be pursued in L[U ] is a question of considerable interest.