Characterizing the powerset by a complete (Scott) sentence

Abstract

This paper is part II of a study on cardinals that are characterizable by a Scott sentence, continuing the work from http://arxiv.org/abs/1007.2426v1. A cardinal $\kappa$ is characterized by a Scott sentence $\phi_M$, if $\phi_M$ has a model of size $\kappa$, but no model of $\kappa^+$. The main question in this paper is the following: Are the characterizable cardinals closed under the powerset operation? We prove that if $\aleph_{\beta}$ is characterized by a Scott sentence, then $2^{\aleph_{\beta+\beta_1}}$ is (homogeneously) characterized by a Scott sentence, for all $0<\beta_1<\omega_1$. So, the answer to the above question is positive, except the case $\beta_1=0$ which remains open. As a consequence we derive that if $\alpha\le\beta$ and $\aleph_{\beta}$ is characterized by a Scott sentence, then $\aleph_{\alpha+\alpha_1}^{\aleph_{\beta+\beta_1}}$ is also characterized by a Scott sentence, for all $\alpha_1<\omega_1$ and $0<\beta_1<\omega_1$. Whence, depending on the model of ZFC, we see that the class of characterizable and homogeneously characterizable cardinals is much richer than previously known. Several open questions are also mentioned at the end.Comment: This paper is an updated version of the second half of version 1 of arXiv:1007.2426v

Hanf numbers and presentation theorems in AECs

“…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). Combining these two with earlier results of Souldatos [Sou13] one obtains several different ways to show the lower bound on the Hanf number for a complete L ω1,ωsentence having maximal models is ω1 . In contrast to [BKS16], all of these examples have no models beyond ω1 .…”

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

“…In [Bau74], Baumgartner used independent families of sets to prove that if κ is homogeneously characterizable, then the same is true for 2 κ . A similar result is Theorem 4.29 of [Sou13] where the assumption of homogeneously characterizability of κ is relaxed to a κ being characterized by a linear order. Given the machinery of homogeneously characterizable cardinals and mergers, our transfer theorem 3.1 has a rather elementary proof.…”

Hanf number for Scott sentences of computable structures

“…Then we inductively extend the definition of I to b = 2 a , where |b| = |a| + 1. The construction is a modified version of that in [12]. We let τ b be the vocabulary τ a ∪ {V, M, U b , P, F, < b }, where V , M , and U b are unary predicates, < b is a binary predicate and F is a ternary predicate.…”

“…Remark. The collection K b described above differs from the collection K(M) in [12] in the following respects:…”

“…Clause (1) follows from the inductive hypothesis. Clause (2) follows from the proofs of Theorems 4.13 and 4.14 in[12] 1 , modified to include U b and P . As in the proof of Lemma 3.2, U D b and P D are defined in the amalgam to be the union of U B b ∪ U C b and P B ∪ P C , respectively.…”