Complete Lω1,ω-sentences with Maximal Models in Multiple Cardinalities

In [BKS14] examples of incomplete sentences are given with maximal models in more than one cardinality. The question was raised whether one can find similar examples of complete sentences. In this paper we give examples of complete Lω1,ωsentences with maximal models in more than one cardinality. From (homogeneous) characterizability of κ we construct sentences with maximal models in κ and in one of κ+, κω , 2κ and more. Indeed, consistently we find sentences with maximal models in uncountably many distinct cardinalities. We unite ideas from [BFKL13, BKL14, Hjo02, Kni77] to find complete sentences of Lω1,ω with maximal models in multiple cardinals. There have been a number of papers finding complete sentences characterizing cardinals beginning with Baumgartner, Malitz and Knight in the 70’s, refined by Laskowski and Shelah in the 90’s and crowned by Hjorth’s characterization of all cardinals below אω1 in the 2002. These results have been refined since. But this is the first paper finding complete sentences with maximal models in two or more cardinals. Our arguments combine and extend the techniques of building atomic models by Fraissé constructions using disjoint amalgamation, pioneered by Laskowski-Shelah and Hjorth, with the notion of homogeneous characterization and tools from Baldwin-Koerwien-Laskowski ([BKL14]). This paper uses specific techniques from [BFKL13, BKL14, Sou14, Sou13] and many proofs are adapted from these sources. We thank the referee for a perceptive and helpful report. Structure of the paper: In Section 1, we explain the merger techniques for combining sentences that homogeneously characterize one cardinal (possibly in terms of another). We adapt the methods of [Sou14] to get a complete sentence with maximal models in κ and κ. In Section 2 we present, for each homogeneously characterizable κ, an Lω1,ω-sentence with maximal models in κ and κ and no larger models. The argument can be generalized to obtain maximal models in κ and κאα , for all countable α. Finally in Section 3, we give various examples of complete sentences with maximal models in a number of cardinalities, modulo appropriate hypotheses on cardinal arithmetic. For example, Corollary 3.2 asserts that if κ is homogeneously characterizable and μ is minimal with 2 ≥ κ there is an Lω1,ω-sentence φκ with maximal models in cardinalities 2 for μ ≤ λ ≤ κ and no models larger than 2. 1. The general construction In this section, for a cardinal κ that admits a homogeneous characterization (Definition1.1), we prove that there exists a complete sentence φκ of Lω1,ω that has maximal models in κ and Date: September 4, 2017. 2010 Mathematics Subject Classification. Primary 03C75, 03C35 Secondary 03C52, 03C30, 03C15.