Abstract We answer a question of Blass, Di Nasso, and Forti [2, 7] by proving, assuming Continuum Hypothesis or Martin’s Axiom, that (1) there exists a P-point which is not interval-to-one and (2) there exists an interval-to-one P-point which is neither quasi-selective nor weakly Ramsey.

Abstract M. Beiglböck, V. Bergelson, and A. Fish proved that if G is a countable amenable group and A and B are subsets of G with positive Banach density, then the product set AB is piecewise syndetic. This means that there is a finite subset E of G such that EAB is thick, that is, EAB contains translates of any finite subset of G . When G = ℤ, this was first proven by R. Jin. We prove a quantitative version of the aforementioned result by providing a lower bound on the density (with respect to a Følner sequence) of the set of witnesses to the thickness of EAB . When G = ℤ d , this result was first proven by the current set of authors using completely different techniques.

Abstract Answers are given to two questions concerning the existence of some sparse subsets of ℋ = {0, 1 … ., H – 1} ⊆ *ℕ. where H is a hyperfinite integer. In §1. we answer a question of Kanovei by showing that for a given cut U in ℋ, there exists a countably determined set X ⊆ ℋ which contains exactly one element in each U-monad, if and only if U = a · ℕ for some a Є ℋ ∖ {0}. In §2, we deal with a question of Keisler and Leth in [6]. We show that there is a cut V ⊆ ℋ such that for any cut U, (i) there exists a U-discrete set X ⊆ ℋ with X + X = ℋ (mod H) provided , (ii) there does not exist any U-discrete set X ⊆ ℋ with X + X = ℋ (mod H) provided . We obtain some partial results for the case U = V.

Abstract We characterize Maharam spectra of Loeb probability spaces and give some applications of the results.

Abstract In this paper we show that the compactness of a Loeb space depends on its cardinality, the nonstandard universe it belongs to and the underlying model of set theory we live in. In §1 we prove that Loeb spaces are compact under various assumptions, and in §2 we prove that Loeb spaces are not compact under various other assumptions. The results in §1 and §2 give a quite complete answer to a question of D.Ross in [9], [11] and [12].

Abstract Type two cuts, bad cuts and very bad cuts are introduced in [10] for studying the relationship between Loeb measure and U-topology of a hyperfinite time line in an ω1-saturated nonstandard universe. The questions concerning the existence of those cuts are asked there. In this paper we answer, fully or partially, some of those questions by showing that: (1) type two cuts exist, (2) the ℵ1-isomorphism property implies that bad cuts exist, but no bad cuts are very bad.

Abstract In § 1 of this paper, we characterize the isomorphism property of nonstandard universes in terms of the realization of some second-order types in model theory. In §2, several applications are given. One of the applications answers a question of D. Ross in [this Journal, vol. 55 (1990), pp. 1233–1242] about infinite Loeb measure spaces.

Abstract In an ω1-saturated nonstandard universe a cut is an initial segment of the hyperinlegers which is closed under addition. Keisler and Leth in [KL] introduced, for each given cut U, a corresponding U-topology on the hyperintegers by letting O be U-open if for any x ϵ O there is a y greater than all the elements in U such that the interval [x − y, x + y] ⊆ O. Let U be a cut in a hyperfinite time line , which is a hyperfinite initial segment of the hyperintegers. U is called a good cut if there exists a U-meager subset of of Loeb measure one. Otherwise U is bad. In this paper we discuss the questions of Keisler and Leth about the existence of bad cuts and related cuts. We show that assuming b > ω1, every hyperfinite time line has a cut with both cofinality and coinitiality uncountable. We construct bad cuts in a nonstandard universe under ZFC. We also give two results about the existence of other kinds of cuts.

Abstract In an ω-saturated nonstandard universe a cut is an initial segment of the hyperintegers which is closed under addition. Keisler and Leth in [KL] introduced, for each given cut U, a corresponding U-topology on the hyperintegers by letting O be U-open if for any x ϵ O there is a y greater than all the elements in U such that the interval [x − y,x + y] ⊆ O. Let U be a cut in a hyperfinite time line , which is a hyperfinite initial segment of the hyperintegers. A subset B of is called a U-Lusin set in if B is uncountable and for any Loeb-Borel U-meager subset X of , B ⋂ X is countable. Here a Loeb-Borel set is an element of the σ-algebra generated by all internal subsets of : In this paper we answer some questions of Keisler and Leth about the existence of U-Lusin sets by proving the following facts. (1) If U = x/N = {y ϵ : ∀n ϵ ℕ(y < x/n)} for some x ϵ , then there exists a U-Lusin set of power κ if and only if there exists a Lusin set of the reals of power κ. (2) If U ≠ x/N but the coinitiality of U is ω, then there are no U-Lusin sets if CH fails. (3) Under ZFC there exists a nonstandard universe in which U-Lusin sets exist for every cut U with uncountable cofinality and coinitiality. (4) In any ω2-saturated nonstandard universe there are no U-Lusin sets for all cuts U except U = x/N.

Abstract In an ω1-saturated nonstandard universe a cut is an initial segment of the hyperintegers which is closed under addition. Keisler and Leth in [KL] introduced, for each given cut U, a corresponding U-topology on the hyperintegers by letting O be U-open if for any x ϵ O there is a y greater than all the elements in U such that the interval [x − y, x + y] ⊆ O. Let U be a cut in a hyperfinite time line , which is a hyperfinite initial segment of the hyperintegers. The U-monad topology of is the quotient topology of the U-topological space modulo U. In this paper we answer a question of Keisler and Leth about the U-monad topologies by showing that when is κ-saturated and has cardinality κ,(1) if the coinitiality of U1, is uncountable, then the U1,-monad topology and the U2-monad topology are homcomorphic iff both U1, and U2 have the same coinitiality; and (2) can produce exactly three different U-monad topologies (up to homeomorphism) for those U's with countable coinitiality. As a corollary can produce exactly four different U-monad topologies if the cardinality of is ω1.

Abstract This paper answers some questions of D. Ross in [R]. In §1, we show that some consequences of the ℵ1- or ℵ1-special model axiom in [R] cannot be proved by the κ-isomorphism property for any cardinal κ. In §2, we show that with one exception, the ℵ0-isomorphism property does imply the remaining consequences of the special model axiom in [R]. In §3, we improve a result in [R] by showing that the κ-special model axiom is equivalent to the ℵ0-special model axiom plus κ-saturation.

Abstract An -structure is called internally presented in a nonstandard universe if its base set and interpretation of every symbol in are internal. A nonstandard universe is said to satisfy the κ-isomorphism property if for any two internally presented -structures and , where has less than κ many symbols, is elementarily equivalent to implies that is isomorphic to . In this paper we prove that the ℵ1-isomorphism property is equivalent to the ℵ0-isomorphism property plus ℵ1-saturation.