arXiv subjects
Publications and source records attributed to Martin Koerwien.
We study the usual notion of Scott rank but in the setting of Polish metric spaces. The signature consists of distance relations: for each rational $q > 0$, there is a relation $R_{<q}(x,y)$ stating that the distance of $x$ and $y $ is less than $q$. We show that compact spaces have Scott rank at most $ω$, and that there are discrete ultrametric spaces of arbitrarily high countable Scott rank.
We introduce the notion of a `pure` Abstract Elementary Class to block trivial counterexamples. We study classes of models of bipartite graphs and show: Main Theorem (cf. Theorem 3.5.2 and Corollary 3.5.6): If $(λ_i : i \le α<\aleph_1)$ is a strictly increasing sequence of characterizable cardinals (Definition 2.1) whose models satisfy JEP$(<λ_0)$, there is an $L_{ω_1,ω}$ -sentence $ψ$ whose models form a pure AEC and (1) The models of $ψ$ satisfy JEP$(<λ_0)$, while JEP fails for all larger cardinals and AP fails in all infinite cardinals. (2) There exist $2^{λ_i^+}$ non-isomorphic maximal models of $ψ$ in $λ_i^+$, for all $i \le α$, but no maximal models in any other cardinality; and (3) $ψ$ has arbitrarily large models. In particular this shows the Hanf number for JEP and the Hanf number for maximality for pure AEC with Lowenheim number $\aleph_0$ are at least $\beth_{ω_1}$. We show that although AP$(κ)$ for each $κ$ implies the full amalgamation property, JEP$(κ)$ for each κdoes not imply the full joint embedding property. We show the main combinatorial device of this paper cannot be used to extend the main theorem to a complete sentence.
We propose a natural theory SO axiomatizing the class of sets of ordinals in a model of ZFC set theory. Both theories possess equal logical strength. Constructibility theory in SO corresponds to a natural recursion theory on ordinals.