Linear orderings and powers of characterizable cardinals

Abstract

The current paper answers an open question of [5]. We say that a countable model M characterizes an infinite cardinal κ, if the Scott sentence of M has a model in cardinality κ, but no models in cardinality κ + . If M is linearly ordered by <, we will say that the linear ordering (M, <) characterizes κ, or that κ is characterizable by (M, <).From [2] we can deduce that if κ is characterizable, then κ + is characterizable by a linear ordering (see theorem 2.4, corollary 2.5). From [5] we know that if κ is characterizable by a dense linear ordering, then 2 κ is characterizable (see theorem 2.7).We show that if κ is homogeneously characterizable (cf. definition 2.2), then κ is characterizable by a dense linear ordering, while the converse fails (theorem 2.3).The main theorems are: 1) If κ > 2 λ is a characterizable cardinal, λ is characterizable by a dense linear ordering and λ is the least cardinal such that κ λ > κ, then κ λ is also characterizable (theorem 5.4), 2) if ℵα and κ ℵα are characterizable cardinals, then the same is true for κ ℵ α+β , for all countable β (theorem 5.5).Combining these two theorems we get that if κ > 2 ℵα is a characterizable cardinal, ℵα is characterizable by a dense linear ordering and ℵα is the least cardinal such that κ ℵα > κ, then for all β < α+ω 1 κ ℵ β is characterizable (theorem 5.7). Also if κ is a characterizable cardinal, then κ ℵα is characterizable, for all countable α (corollary 5.6). Structure of the paperThroughout the whole paper we work with countable languages L and when we refer to a dense linear ordering we mean a dense linear ordering without endpoints. The first two sections provide some background material for the characterizable cardinals and for the dense linear orderings respectively. Section 4 contains the construction that proves the following Theorem 1.1. If κ is a characterizable cardinal, then κ ℵ1 is also a characterizable cardinal.This appears as theorem 4.18 in section 4 and it will be easily generalized to λ ≥ ℵ 1 in the last section. Characterizable cardinalsThis section provides the necessary background on characterizable cardinals.

Notes on Cardinals That Are Characterizable by a Complete (Scott) Sentence

“…This section contains the basic definitions and background theorems. A similar discussion also appears in [9].…”

“…">Introduction and known resultsThis section contains the basic definitions and background theorems. A similar discussion also appears in [9].Let the signature of our logic be L. We will consider only countable L. For basic definitions in infinitary logic L ω1,ω , the reader can refer to [4]. In L ω1,ω we allow formulas that have negation, universal/existential quantification and countably long disjunctions and conjunctions, but not countably long quantification.…”

Complete Lω1,ω‐sentences with maximal models in multiple cardinalities

“…1 1) Baumgartner; see also Theorem 3.4 of [Sou13]; 2) Theorem 3.6, [Sou14]; 3)Corollary 5.6, [Sou12].…”

“…Replacing the construction that characterizes κ ω from [Sou14] with the construction that characterized κ ℵα , α < ω 1 , from [Sou12] (cf. Theorem 3) one can prove the following theorem.…”

“…Particular examples of homogeneously characterizable cardinals are given by [Bau74], [Hjo02], [Sou12], [Sou13], [Sou14], [BKL14].…”