Currently the two popular ways to practice Robinson’s nonstandard analysis are the model-theoretic approach and the axiomatic/syntactic approach. It is sometimes claimed that the internal axiomatic approach is unable to handle constructions relying on external sets. We show that internal frameworks provide successful accounts of nonstandard hulls and Loeb measures. The basic fact this work relies on is that the ultrapower of the standard universe by a standard ultrafilter is naturally isomorphic to a subuniverse of the internal universe.

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)

It is a theorem of Prikry [7] that ifκcarries a uniformη-descendingly complete ultrafilter then the stationary reflection propertyfails. In this paper we will derive similar results, but here from properties of filters rather than ultrafilters.Throughoutκandηwill denote regular cardinals withη<κ, andIwill denote an ideal onκ, by which we mean a setI⊆P such that Iis closed under taking subsets and finite unions and αЄIfor eachα<κ, butκ∉I.Iis said to beμ-complete if it is closed under taking unions of size <μ,I* = {X⊆κ∣κ−XЄI} is (...) the filter dual toIand ifAЄI+ −I), thenI∣Ais the ideal onκgiven byI∣A= {X⊆κ∣X∩AЄI}. Ifh: A→κthenhis said to be unbounded modIif for eachα<κ,h−1 = {ξЄA∣h<α} ЄIand a least function forIifhis unbounded modIand wheneverg: A→κis a function, unbounded modI, then {ξЄA∣g{ξ) shrink)