Extending the Language of Set Theory

语言片段力迫、扩展Namba力迫及解释中带约束的理论模型

Forcing with Language Fragments, Extending Namba Forcing, and Models of Theories with Constraints in Interpretation

本文提出了一种基于语言片段融合的力迫框架，并应用于扩展Namba问题变体及解释中带约束理论（TCIs）的模型分析，同时奠定了TCIs理论的基础。

Extending the extensional level of the Minimalist Foundation to axiomatic set theories

We introduce extensions by rules of the extensional level of the Minimalist Foundation which turn out to be equivalent to constructive and classical axiomatic set theories.

Extending Properly n-REA Sets

In [5] Soare and Stob prove that if $A$ is an r.e. set which isn't computable then there is a set of the form $A \oplus W^A_e$ which isn't of r.e. Turing degree. If we define a properly $n+1$-REA set to be an $n+1$-REA set which isn't Turing equivalent to any $n$-REA set this result shows that every properly 1-REA set can be extended to a properly 2-REA set. This result was extended in [1] by Cholak and Hinman who proved that every 2-REA set can be extended to a properly 3-REA set. This leads naturally to the hypothesis that every properly $n$-REA set can be extended to a properly $n+1$-REA se…

End extending models of set theory via power admissible covers

Motivated by problems involving end extensions of models of set theory, we develop the rudiments of the power admissible cover construction (over ill-founded models of set theory), an extension of the machinery of admissible covers invented by Barwise as a versatile tool for generalizing model-theoretic results about countable well-founded models of set theory to countable ill-founded ones. Our development of the power admissible machinery allows us to obtain new results concerning powerset-preserving end extensions and rank extensions of countable models of subsystems of $\mathsf{ZFC}$. The c…