On properties of theories which preclude the existence of universal models

Universal theories and compactly expandable models

Our aim is to solve a quite old question on the difference between expandability and compact expandability. Toward this, we further investigate the logic of countable cofinality.

域的普遍-存在理论

Universal-existential theories of fields

本文研究了域的普遍-存在理论，探讨了不同域结构之间的普遍-存在片段关系及一一还原。

The universality spectrum of stable unsuperstable theories

It is shown that if T is stable unsuperstable, and aleph_1< lambda =cf(lambda)< 2^{aleph_0}, or 2^{aleph_0} < mu^+< lambda =cf(lambda)< mu^{aleph_0} then T has no universal model in cardinality lambda, and if e.g. aleph_omega < 2^{aleph_0} then T has no universal model in aleph_omega. These results are generalized to kappa =cf(kappa) < kappa (T) in the place of aleph_0. Also: if there is a universal model in lambda >|T|, T stable and kappa < kappa (T) then there is a universal tree of height kappa +1 in cardinality lambda .

Counting siblings in universal theories

We show that if a countable structure $M$ in a finite relational language is not cellular, then there is an age-preserving $N \supseteq M$ such that $2^{\aleph_0}$ many structures are bi-embeddable with $N$. The proof proceeds by a case division based on mutual algebraicity.

Universal theories categorical in power and kappa-generated models

We investigate a notion called uniqueness in power kappa that is akin to categoricity in power kappa, but is based on the cardinality of the generating sets of models instead of on the cardinality of their universes. The notion is quite useful for formulating categoricity-like questions regarding powers below the cardinality of a theory. We prove, for (uncountable) universal theories T, that if T is kappa-unique for one uncountable kappa, then it is kappa-unique for every uncountable kappa ; in particular, it is categorical in powers greater than the cardinality of T.

真理的普遍性质

Universal properties of truth

本文研究了在子语言中可定义真理或满足谓词的模型的抽象模型论性质，分析了确保真理谓词可定义性的条件，并探讨了相关公理化理论及其相互可定义性关系。