Abstract

Some highly saturated models of Peano Arithmetic are constructed in this paper, which consists of two independent sections. In § 1 we answer a question raised in [10] by constructing some highly saturated, rather classless models of PA. A question raised in [7], [3], ]4] is answered in §2, where highly saturated, nonstandard universes having no bad cuts are constructed.Highly saturated, rather classless models of Peano Arithmetic were constructed in [10]. The main result proved there is the following theorem. If λ is a regular cardinal andis aλ-saturated model of PA such that ∣M∣ >λ, thenhas an elementary extensionof the same cardinality which is alsoλ-saturated and which, in addition, is rather classless. The construction in [10] produced a modelfor which cf() =λ+. We asked in Question 5.1 of [10] what other cofinalities could such a modelhave. This question is answered here in Theorem 1.1 of §1 by showing that any cofinality not immediately excluded is possible. Its proof does not depend on the theorem from [10]; in fact, the proof presented here gives a proof of that theorem which is much simpler and shorter than the one in [10].Recursively saturated, rather classless κ-like models of PA were constructed in [9]. In the case of singular κ such models were constructed whenever cf(κ) > ℵ0; no additional set-theoretic hypothesis was needed.