Type Two Cuts, Bad Cuts and Very Bad Cuts

Abstract

Type two cuts, bad cuts and very bad cuts are introduced by Keisler and Leth 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 in any nonstandard model of Peano Arithmetic, (2) the $\aleph_1$-isomorphism property implies that bad cuts exist, but no bad cuts are very bad.