A construction of the real number system based on almost homomorphisms of the integers $\mathbb {Z}$ was proposed by Schanuel, Arthan, and others. We combine such a construction with the ultrapower or limit ultrapower construction to construct the hyperreals out of integers. In fact, any hyperreal field, whose universe is a set, can be obtained by such a one-step construction directly out of integers. Even the maximal (i.e., On -saturated) hyperreal number system described by Kanovei and Reeken (2004) and independently (...) by Ehrlich (2012) can be obtained in this fashion, albeit not in NBG . In NBG , it can be obtained via a one-step construction by means of a definable ultrapower (modulo a suitable definable class ultrafilter). (shrink)

This paper answers some questions of D. Ross in [R]. In § 1, we show that some consequences of the ℵ0- 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. (shrink)

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

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

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.

Three results in [14] and one in [8] are analyzed in Sections 3–6 in order to supply examples on Loeb probability spaces, which distinguish the different strength among three generalizations of k-saturation, as well to answer some questions in Section 7 of [15]. In Section 3 we show that not every automorphism of a Loeb algebra is induced by an internal permutation, in Section 4 we show that if the 1-special model axiom is true, then every automorphism of a Loeb (...) algebra is induced by a point-automorphism, in Section 5 we show that not every measure-preserving homomorphism from a small subalgebra to a Loeb algebra is induced by an internal permutation, without assuming full-saturation, in Section 6 we show that, under some cardinality assumptions, the 1-isomorphism property does not guarantee the compactness of a Loeb space, and in Section 7 an application of the 1-special model axiom is given on the existence of ergodic transformations of a Loeb space, which partially answers Problem 2.3 of [5]. (shrink)

We prove that if is a model of size at most [kappa], λ[kappa] = λ, and a game sentence of length 2λ is true in a 2λ-saturated model ≡ , then player has a winning strategy for a related game in some ultrapower ΠD of . The moves in the new game are taken in the cartesian power λA, and the ultrafilter D over λ must be chosen after the game is played. By taking advantage of the expressive power of (...) game sentences, we obtain several applications showing the existence of ultrapowers with certain properties. In each case, it was previously known that such ultrapowers exist under the assumption of the GCH, and we get them without the GCH. Article O. (shrink)

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 ω 1 -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 ℵ 1 -isomorphism property implies that bad cuts exist, but no bad cuts are (...) very bad. (shrink)

In this paper we probe the possibilities of creating a Kurepa tree in a generic extension of a ground model of CH plus no Kurepa trees by an ω1-preserving forcing notion of size at most ω1. In Section 1 we show that in the Lévy model obtained by collapsing all cardinals between ω1 and a strongly inaccessible cardinal by forcing with a countable support Lévy collapsing order, many ω1-preserving forcing notions of size at most ω1 including all ω-proper forcing notions (...) and some proper but not ω-proper forcing notions of size at most ω1 do not create Kurepa trees. In Section 2 we construct a model of CH plus no Kurepa trees, in which there is an ω-distributive Aronszajn tree such that forcing with that Aronszajn tree does create a Kurepa tree in the generic extension. At the end of the paper we ask three questions. (shrink)

Let $\omega$ be the first infinite ordinal (or the set of all natural numbers) with the usual order $<$ . In § 1 we show that, assuming the consistency of a supercompact cardinal, there may exist an ultrapower of $\omega$ , whose cardinality is (1) a singular strong limit cardinal, (2) a strongly inaccessible cardinal. This answers two questions in [1], modulo the assumption of supercompactness. In § 2 we construct several $\lambda$ -Archimedean ultrapowers of $\omega$ under some large cardinal (...) assumptions. For example, we show that, assuming the consistency of a measurable cardinal, there may exist a $\lambda$ -Archimedean ultrapower of $\omega$ for some uncountable cardinal $\lambda$ . This answers a question in [8], modulo the assumption of measurability. (shrink)

By an ω1 we mean a tree of power ω1 and height ω1. An ω1-tree is called a Kurepa tree if all its levels are countable and it has more than ω1 branches. An ω1-tree is called a Jech-Kunen tree if it has κ branches for some κ strictly between ω1 and $2^{\omega _1 }$ . In Sect. 1, we construct a model ofCH plus $2^{\omega _1 } > \omega _2$ , in which there exists a Kurepa tree with not (...) Jech-Kunen subtrees and there exists a Jech-Kunen tree with no Kurepa subtrees. This improves two results in [Ji1] by not only eliminating the large cardinal assumption for [Ji1, Theorem 2] but also handling two consistency proofs of [Ji1, Theorem 2 and Theorem 3] simultaneously. In Sect. 2, we first prove a lemma saying that anAxiom A focing of size ω1 over Silver's model will not produce a Kurepa tree in the extension, and then we apply this lemma to prove that, in the model constructed for Theorem 2 in [Ji1], there exists a Jech-Kunen tree and there are no Kurepa trees. (shrink)

A P-point ultrafilter over \(\omega \) is called an interval P-point if for every function from \(\omega \) to \(\omega \) there exists a set _A_ in this ultrafilter such that the restriction of the function to _A_ is either a constant function or an interval-to-one function. In this paper we prove the following results. (1) Interval P-points are not isomorphism invariant under \(\textsf{CH}\) or \(\textsf{MA}\). (2) We identify a cardinal invariant \(\textbf{non}^{**}({\mathcal {I}}_{\tiny {\hbox {int}}})\) such that every filter base (...) of size less than continuum can be extended to an interval P-point if and only if \(\textbf{non}^{**}({\mathcal {I}}_{\tiny {\hbox {int}}})={\mathfrak {c}}\). (3) We prove the generic existence of slow/rapid non-interval P-points and slow/rapid interval P-points which are neither quasi-selective nor weakly Ramsey under the assumption \({\mathfrak {d}}={\mathfrak {c}}\) or \(\textbf{cov}({\mathcal {B}})={\mathfrak {c}}\). (shrink)

This paper reports recent progress in applying nonstandard analysis to additive number theory, especially to problems involving upper Banach density.

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 ∈ O there is a y greater than all the elements in U such that the interval $\lbrack x - y, x + y\rbrack \subseteq O$ . Let U be a cut in a hyperfinite time (...) line H, which is a hyperfinite initial segment of the hyperintegers. U is called a good cut if there exists a U-meager subset of H 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 $\mathbf{b} > \omega_1$, 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. (shrink)

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 $\S1$ we prove that Loeb spaces are compact under various assumptions, and in $\S2$ we prove that Loeb spaces are not compact under various other assumptions. The results in $\S1$ and $\S2$ give a quite complete answer to a question of D. Ross in [9], [11] and [12].

By an ω1-tree we mean a tree of cardinality ω1 and height ω1. An ω1-tree is called a Kurepa tree if all its levels are countable and it has more than ω1 branches. An ω1-tree is called a Jech–Kunen tree if it has κ branches for some κ strictly between ω1 and 2ω1. A Kurepa tree is called an essential Kurepa tree if it contains no Jech–Kunen subtrees. A Jech–Kunen tree is called an essential Jech–Kunen tree if it is no (...) Kurepa subtrees. In this paper we prove that it is consistent with CH and 2ω1 #62; ω2 that there exist essential Kurepa trees and there are no essential Jech–Kunen trees, it is consistent with CH and 2ω1 #62; ω2 plus the existence of a Kurepa tree with 2ω1 branches that there exist essential Jech–Kunen trees and there are no essential Kurepa trees. In the second result we require the existence of a Kurepa tree with 2ω1 branches in order to avoid triviality. (shrink)

Answers are given to two questions concerning the existence of some sparse subsets of $\mathscr{H} = \{0, 1,..., H - 1\} \subseteq * \mathbb{N}$ , 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 $X \subseteq \mathscr{H}$ which contains exactly one element in each U-monad, if and only if U = a · N for some $a \in \mathscr{H} (...) \backslash \{0\}$ . In §2, we deal with a question of Keisler and Leth in [6]. We show that there is a cut $V \subseteq \mathscr{H}$ such that for any cut U, (i) there exists a U-discrete set $X \subseteq \mathscr{H}$ with X + X = H (mod H) provided $U \subsetneqq V$ , (ii) there does not exist any U-discrete set $X \subseteq \mathscr{H}$ with X + X = H (mod H) provided $\supsetneqq V$ . We obtain some partial results for the case U = V. (shrink)

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

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

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 ∈ O there is a y greater than all the elements in U such that the interval $\lbrack x - y, x + y\rbrack \subseteq O$ . Let U be a cut in a hyperfinite time (...) line H, which is a hyperfinite initial segment of the hyperintegers. A subset B of H is called a U-Lusin set in H if B is uncountable and for any Loeb-Borel U-meager subset X of H, B ∩ X is countable. Here a Loeb-Borel set is an element of the σ-algebra generated by all internal subsets of H. 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 $U = x/\mathbb{N} = \{y \in \mathscr{H}: \forall n \in \mathbb{N}(y < x/n)\}$ for some x ∈ H, 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. (shrink)

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 ∈ O there is a y greater than all the elements in U such that the interval $\lbrack x - y, x + y\rbrack \subseteq O$ . Let U be a cut in a hyperfinite time (...) line H, which is a hyperfinite initial segment of the hyperintegers. The U-monad topology of H is the quotient topology of the U-topological space H modulo U. In this paper we answer a question of Keisler and Leth about the U-monad topologies by showing that when H 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 homeomorphic iff both U1 and U2 have the same coinitiality; and (2) H can produce exactly three different U-monad topologies (up to homeomorphism) for those U's with countable coinitiality. As a corollary H can produce exactly four different U-monad topologies if the cardinality of H is ω1. (shrink)

Let U be an initial segment of $^*{\mathbb N}$ closed under addition (such U is called a cut) with uncountable cofinality and A be a subset of U, which is the intersection of U and an internal subset of $^*{\mathbb N}$ . Suppose A has lower U-density α strictly between 0 and 3/5. We show that either there exists a standard real $\epsilon$ > 0 and there are sufficiently large x in A such that | (A+A) ∩ [0, 2x]| > (...) (10/3+ $\epsilon$ ) | A ∩ [0, x]| or A is a large subset of an arithmetic progression of difference greater than 1 or A is a large subset of the union of two arithmetic progressions with the same difference greater than 2 or A is a large subset of the union of three arithmetic progressions with the same difference greater than 4. (shrink)