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

Abstract

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 Lω 1 ,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 + , κ ω , 2 κ 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 L ω1,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 ℵ ω1 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. All models of these sentences have cardinality less than ω1 .

Disjoint Amalgamation in Locally Finite Aec

“…They also asserted that they had a similar construction to characterize all cardinals up to ℵ ω , but there was a mistake in the proof. 8 We remedy that now.…”

“…The examples of Kolesnikov and Lambie-Hanson are distinctive as amalgamation is equivalent to disjoint amalgamation: some results depend on a generalized Martin axiom. The construction of non-trivial spectra of disjoint embedding [6] and of maximal models for complete sentences [8] rely on the current paper.…”

“…In [3] it was shown that this induced class satisfies JEP and (< ℵ 0 , 2)-disjoint amalgamation; hence has a generic model M h ; and that V (M h ) is an infinite set of absolute indiscernibles. As in [3], one constructs further examples by 'merging' two theories on the set of absolute indiscernibles; applications requiring the ability to fix the cardinality of the set of absolute indiscernibles appear in [4,8]. We say N is an (ℵ r , ℵ k )-model if N = ℵ r and |V (N )| = ℵ k .…”

“…If (K 0 , ≤) satisfies our notion of disjoint (< ℵ 0 , r + 1) amalgamation then (by a new construction) bothK and R have models in ℵ r and satisfy disjoint (≤ ℵ s , r − s) amalgamation for s ≤ r. We modify [3] to show the existence of homogeneous characterizations (Definition 2.27) (arising from [11]); this leads to new examples of joint embedding spectra in [4,8]. For finer control of the uncountable models, especially of rich models, we introduce the notion of frugal amalgamation (Definition 2.29); the amalgam must have universe the union of the constituent models.…”

Hanf numbers and presentation theorems in AECs

“…Again, the result is considerably more complicated for complete sentences. But [BS15b] show that there is a sentence φ in a vocabulary with a predicate X such that if M |= φ, |M | ≤ |X(M )| + and for every κ there is a model with |M | = κ + and |X(M )| = κ. Further they note that if there is a sentence φ that homogenously characterizes κ, then there is a sentence φ ′ with a new predicate B such that φ ′ also characterizes κ, B defines a set of absolute indiscernibles in the countable model, and there are models M λ for λ ≤ κ such that (|M |, |B(M λ )|) = (κ, λ).…”