It is proved here, assuming Projective Determinacy, that every ascending sequence of -degrees has a minimal strict upper bound but no least strict upper bound. This generalizes a result of Friedman for n = 1.

Our general notation and terminology will be that of [Ke1] and [Mo1]. Letters i,j, k,… denote members of ω and α, β, ϒ,… members of ωω i.e. reals. Projective Determinacy (PD) is the hypothesis that every projective set of reals in determined, while in general for a collection of sets of reals Γ, Determinacy (Γ) abbreviates the statement that every set in Γ is determined.

§1. -degrees. For each m ≥ 1 and α, β Є ωω let α ≤mβ ⇔ α Є (β), α <mβ ⇔ α ≤mβ ∧ β ≰mα, and α ≡mβ ⇔ α ≤mβ ∧ β ≤mα.Clearly ≡m is an equivalence relation on ωω. The ≡m -equivalence class of α Є ωω is called its -degree, in symbols

If d = [α]m, e = [β]m then we define

One of the main results of Barwise [2] (see also [7, Chapter VIII]) showed that the s – reflection principle for a set A is equivalent to Σ1-compactness of . Here A is any transitive p.r. closed set, and is the infinitary language on A which allows conjunction and disjunction over arbitrary sets Φ Є A, and finite quantification.

In this paper we consider languages , where B is a Δ0 subset of A, which is like but we allow quantifiers ∀x and ∃x where x is any set of variables indexed by an element of B. A treatment similar to that of [2] for establishes a sufficient, and in some cases necessary, condition for to be Σ1-compact. The use of infinitary Skolem functions is intrinsic to the method, so to avoid a separate development of the rudiments of the Skolem language we actually define to have b-ary relation and function symbols for every b Є B.

We characterize the model companions of universal Horn classes generated by a two-element algebra (or ordered two-element algebra). We begin by proving that given two mutually model consistent classes M and N of (respectively ) structures, with , , provided that an -definability condition for the function and relation symbols of holds. We use this, together with Post's characterization of ISP(A), where A is a two-element algebra, to show that the model companions of these classes essentially lie in the classes of posets and semilattices, or characteristic two groups and relatively complemented distributive lattices.

Forcing with [Κ]κ over a model of set theory with a strong partition cardinal, M. Spector produced a generic ultrafilter G on κ such that κκ/G is not well-founded.

Theorem. Let G be Spector-generic over a model M of , for all α < κ.

1) Every cardinal (well-ordered or not) of M is a cardinal of M[G].

2) If A ∈ M[G] is a well-ordered subset of M, then Ae M. Let Φ = Κκ/G.

3) There is an ultrafilter U on Φ such that every member of U has a subset of type Φ, and the intersection of any well-ordered subset of U is in U.

4) Φ satisfies for all α <ℵ1 and all ordinals β.

5) There is a linear order Φ′ with property 3) above which is not “weakly compact”, i.e., Φ′ ↛ (Φ′)2.

We show that after forcing with a countable support iteration or a finite product of Sacks or splitting forcing over L, every analytic hypergraph on a Polish space admits a $\mathbf {\Delta }^1_2$ maximal independent set. This extends an earlier result by Schrittesser (see [25]). As a main application we get the consistency of $\mathfrak {r} = \mathfrak {u} = \mathfrak {i} = \omega _2$ together with the existence of a $\Delta ^1_2$ ultrafilter, a $\Pi ^1_1$ maximal independent family, and a $\Delta ^1_2$ Hamel basis. This solves open problems of Brendle, Fischer, and Khomskii [5] and the author [23]. We also show in ZFC that $\mathfrak {d} \leq \mathfrak {i}_{cl}$ , addressing another question from [5].

The probability logic is a logic with a natural interpretation on probability spaces (thus, a logic whose model theory is part of probability theory rather than a system for putting probabilities on formulas of first order logic). Its exact definition and basic development are contained in the paper [3] of H. J. Keisler and the papers [1] and [2] of the author. Building on work in [2], we prove in this paper the following probabilistic interpolation theorem for .

Let L be a countable relational language, and let A be a countable admissible set with ω ∈ A (in this paper some probabilistic notation will be used, but ω will always mean the least infinite ordinal). is the admissible fragment of corresponding to A. We will assume that L is a countable set in A, as is usual in practice, though all that is in fact needed for our proof is that L be a set in A which is wellordered in A.

Theorem. Let ϕ(x) and ψ(x) be formulas of LAP such that

where ε ∈ [0, 1) is a real in A (reals may be defined in the usual way as Dedekind cuts in the rationals). Then for any real d > ε¼, there is a formula θ(x) of (L(ϕ) ∩ L(ψ))AP such that

and

By taking a closer look at the construction of an Ackermann function we see that between any primitive recursive degree and its Ackermann modification there is a dense chain of primitive recursive degrees.

We prove that the only implications between four notions for Σ½-sets of reals are

We study the following conjecture.

Conjecture.Let T be an ω-stable theory with continuum many countable models. Then either i) T has continuum many complete extensions in L 1(T), or ii) some complete extension of T in L 1has continuum many L 1-types without parameters.

By Shelah's proof of Vaught's conjecture for ω-stable theories, we know that there are seven types of ω-stable theory with continuum many countable models. We show that the conjecture is true for all but one of these seven cases. In the last case we show the existence of continuum many L 2-types.