ﻻ يوجد ملخص باللغة العربية
We use set-theoretic tools to make a model-theoretic contribution. In particular, we construct a emph{single} $mathcal{L}_{omega_1,omega}$-sentence $psi$ that codes Kurepa trees to prove the consistency of the following: (1) The spectrum of $psi$ is consistently equal to $[aleph_0,aleph_{omega_1}]$ and also consistently equal to $[aleph_0,2^{aleph_1})$, where $2^{aleph_1}$ is weakly inaccessible. (2) The amalgamation spectrum of $psi$ is consistently equal to $[aleph_1,aleph_{omega_1}]$ and $[aleph_1,2^{aleph_1})$, where again $2^{aleph_1}$ is weakly inaccessible. This is the first example of an $mathcal{L}_{omega_1,omega}$-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, $psi$ 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) $2^{aleph_0}<aleph_{omega_1}<2^{aleph_1}$ and there exists an $mathcal{L}_{omega_1,omega}$-sentence with models in $aleph_{omega_1}$, but no models in $2^{aleph_1}$. This relates to a conjecture by Shelah that if $aleph_{omega_1}<2^{aleph_0}$, then any $mathcal{L}_{omega_1,omega}$-sentence with a model of size $aleph_{omega_1}$ also has a model of size $2^{aleph_0}$. Our result proves that $2^{aleph_0}$ can not be replaced by $2^{aleph_1}$, even if $2^{aleph_0}<aleph_{omega_1}$.
In partial answer to a question posed by Arnie Miller (http://www.math.wisc.edu/~miller/res/problem.pdf) and X. Caicedo, we obtain sufficient conditions for an L_{omega_1,omega} theory to have an independent axiomatization. As a consequence we obtain
We obtain a computable structure of Scott rank omega_1^{CK} (call this ock), and give a general coding procedure that transforms any hyperarithmetical structure A into a computable structure A such that the rank of A is ock, ock+1, or < ock iff the same is true of A.
We give Scott sentences for certain computable groups, and we use index set calculations as a way of checking that our Scott sentences are as simple as possible. We consider finitely generated groups and torsion-free abelian groups of finite rank. Fo
The Hanf number for a set $S$ of sentences in $L_{omega_1,omega}$ (or some other logic) is the least infinite cardinal $kappa$ such that for all $varphiin S$, if $varphi$ has models in all infinite cardinalities less than $kappa$, then it has models