Kurepa trees and spectra of $\mathcal{L}_{ω_1,ω}$-sentences

Abstract

We construct a single Lω 1 ,ω -sentence ψ that codes Kurepa trees to prove the consistency of the following:(1) The spectrum of ψ is consistently equal to [ℵ 0 , ℵω 1 ] and also consistently equal to [ℵ 0 , 2 ℵ 1 ), where 2 ℵ 1 is weakly inaccessible.(2) The amalgamation spectrum of ψ is consistently equal to, where again 2 ℵ 1 is weakly inaccessible. This is the first example of an Lω 1 ,ω -sentence whose spectrum and amalgamation spectrum are consistently both right-open and right-closed. It also provides a positive answer to a question in [18]. (3) Consistently, ψ has maximal models in finite, countable, and uncountable many cardinalities. This complements the examples given in [1] and [2]of sentences with maximal models in countably many cardinalities. (4) 2 ℵ 0 < ℵω 1 < 2 ℵ 1 and there exists an Lω 1 ,ω -sentence with models in ℵω 1 , but no models in 2 ℵ 1 . This relates to a conjecture by Shelah that if ℵω 1 < 2 ℵ 0 , then any Lω 1 ,ωsentence with a model of size ℵω 1 also has a model of size 2 ℵ 0 . Our result proves that 2 ℵ 0 can not be replaced by 2 ℵ 1 , even if 2 ℵ 0 < ℵω 1 .