We prove quantifier elimination for the field ℚp((tℚ)) (the completion of the field of Puiseux series over ℚp) in Macintyre's language together with symbols for functions in a class containing both t-adically and p-adically overconvergent functions. We also show that the theory of ℚp((tℚ)) is b-minimal in this language.

There exists a family of sets of countable ordinals such that:

(1) max Bα = α,

(2) if α ∈ Bβ then Bα ⊆ Bβ,

(3) if λ ≤ α and λ is a limit ordinal then Bα ∩ λ is not in the ideal generated by the Bβ, β < α, and by the bounded subsets of λ,

(4) there is a partition of ω1 such that for every α and every n, Bα ∩ An is finite.

The game is played on a complete Boolean algebra , by two players. White and Black, in κ-many moves (where κ is an infinite cardinal). At the beginning White chooses a non-zero element p ∈ . In the α-th move White chooses pα ∈ (0, p) and Black responds choosing iα ∈{0, 1}. White winsthe play iff . where and .

The corresponding game theoretic properties of c.B.a.'s are investigated. So, Black has a winning strategy (w.s.) if κ ≥ π() or if contains a κ-closed dense subset. On the other hand, if White has a w.s., then κ ∈ . The existence of w.s. is characterized in a combinatorial way and in terms of forcing. In particular, if 2<κ = κ ∈ Reg and forcing by preserves the regularity of κ, then White has a w.s. iff the power 2κ is collapsed to κ in some extension. It is shown that, under the GCH, for each set S ⊆ Reg there is a c.B.a. such that White (respectively. Black) has a w.s. for each infinite cardinal κ ∈ S (resp. κ ∉ S). Also it is shown consistent that for each κ ∈ Reg there is a c.B.a. on which the game is undetermined.

Let σ() denote the number of automorphisms of a model of power ω1. We derive a necessary and sufficient condition in terms of trees for the existence of an with . We study the sufficiency of some conditions for . These conditions are analogous to conditions studied by D. Kueker in connection with countable models.

In [3] Myhill has constructed a complete system K which allows in it the development of a large and important section of classical mathematics. Completeness is achieved essentially by sacrificing universal quantification and introducing instead the proper ancestral as a primitive idea.

In the following we are presenting a system K0 which will be shown to be equivalent to K (i.e. the primitive operators of both systems are mutually definable in terms of one another). K0 is also complete and covers the same ground as K. K0, however, differs from K by the introduction of the limited universal quantifier instead of the proper ancestral and of concatenation instead of the ordered-pair function as primitive operators. By a further reduction K0 will be shown to be equivalent to the system K1 not containing the abstraction-operator and the class-membership relation.

Let p be a set. A function Φ is uniformly Σ 1(p) in every admissible set if there is a Σ 1 formula ϕ in the parameter p so that ϕ defines Φ in every Σ 1-admissible set which includes p. A theorem of Van de Wiele states that if Φ is a total function from sets to sets then Φ is uniformly Σ 1 in every admissible set if and only if it is E-recursive. A function is ESp -recursive if it can be generated from the schemes for E-recursion together with a selection scheme over the transitive closure of p. The selection scheme is exactly what is needed to insure that the ESP -recursively enumerable predicates are closed under existential quantification over the transitive closure of p. Two theorems are established: a) If the transitive closure of p is countable then a total function on sets is ESp -recursive if and only if it is uniformly Σ 1(p) in every admissible set. b) For any p, if Φ is a function on the ordinal numbers then Φ is ESP -recursive if and only if it is uniformly Σ 1(p) in every admissible set.

Where AR is the set of arithmetic Turing degrees, 0(ω) is the least member of {a(2) ∣ a is an upper bound on AR}. This situation is quite different if we examine HYP, the set of hyperarithmetic degrees. We shall prove (Corollary 1) that there is an a, an upper bound on HYP, whose hyperjump is the degree of Kleene's . This paper generalizes this example, using an iteration of the jump operation into the transfinite which is based on results of Jensen and is detailed in [3] and [4]. In § 1 we review the basic definitions from [3] which are needed to state the general results.

In [O1], we gave algebraic characterizations of elementary equivalence for finitely generated finite-by-abelian groups, i.e. finitely generated FC-groups. We also provided several examples of finitely generated finite-by-abelian groups which are elementarily equivalent without being isomorphic.

In this paper, we shall use our previous results to describe precisely the models of the theories of finitely generated finite-by-abelian groups and the elementary embeddings between these models.

Ziegler studies in [2] the expressive power of (Lωω)t for T3 topological spaces. He defines for every natural number n the set of n-types by induction: . If A is a T3 space, the n-type of a ∈ A is defined inductively by: : in every neighborhood of a there is an a′ ≠ a with tn(a′, A) = α}. These types are (Lωω)t-definable. Then, it is shown that two T3 spaces are (Lωω)t-equivalent precisely if for every n-type α they have the same number of points with n-type α (cf. [2]).

In order to study the expressability of (Lω1ω)t, for T3 spaces, we introduce in this paper the notion of accessible set. Looking at the behaviour of convergence by means of this notion, we refine Ziegler's notion of n-type and introduce a new set Sn of n-types, which are (Lω1ω)t-definable. Then, we prove a characterization of (Lω1ω)t-equivalence for a wide class of T3 spaces. A T3 space A belongs to this class if there is a κ ∈ ω such that, for every n ∈ ω, there are at most κ n-types in Sn which are satisfiable in A. Such a space is said to be of a-finite type. Some relations between these spaces and the spaces of finite type in the sense of [2] are shown in the last section.

The contents of the present paper are treated in more detail in [3].