AN AXIOMATIC APPROACH TO FORCING IN A GENERAL SETTING

R. A. Freire, P. Holy
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引用次数: 0

Abstract

Abstract The technique of forcing is almost ubiquitous in set theory, and it seems to be based on technicalities like the concepts of genericity, forcing names and their evaluations, and on the recursively defined forcing predicates, the definition of which is particularly intricate for the basic case of atomic first order formulas. In his [3], the first author has provided an axiomatic framework for set forcing over models of $\mathrm {ZFC}$ that is a collection of guiding principles for extensions over which one still has control from the ground model, and has shown that these axiomatics necessarily lead to the usual concepts of genericity and of forcing extensions, and also that one can infer from them the usual recursive definition of forcing predicates. In this paper, we present a more general such approach, covering both class forcing and set forcing, over various base theories, and we provide additional details regarding the formal setting that was outlined in [3].
在一般情况下对强迫的一种公理化的方法
强迫技术在集合理论中几乎无处不在,它似乎是基于泛型概念、强迫名称及其求值等技术,以及递归定义的强迫谓词,其定义对于原子一阶公式的基本情况尤其复杂。在他的[3]中,第一作者为$\ mathm {ZFC}$模型上的集合强迫提供了一个公理框架,该框架是一组仍然可以从基础模型控制的扩展指导原则的集合,并表明这些公理必然导致一般的泛型和强迫扩展的概念,并且人们可以从中推断出强迫谓词的通常递归定义。在本文中,我们提出了一种更一般的方法,涵盖了各种基本理论的类强制和集强制,并提供了关于[3]中概述的正式设置的额外细节。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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