ω-Models of finite set theory

Logic group preprint series Pub Date : 2008-05-29 DOI:10.1017/CBO9780511910616.004

A. Enayat, J. Schmerl, A. Visser

引用次数: 13

Abstract

Finite set theory, here denoted ZFfin, is the theory ob- tained by replacing the axiom of infinity by its negation in the usual axiomatization of ZF (Zermelo-Fraenkel set theory). An !-model of ZFfin is a model in which every set has at most finitely many elements (as viewed externally). Mancini and Zambella (2001) em- ployed the Bernays-Rieger method of permutations to construct a recursive !-model of ZFfin that is nonstandard (i.e., not isomor- phic to the hereditarily finite sets V!). In this paper we initiate the metamathematical investigation of !-models of ZFfin. In par- ticular, we present a new method for building !-models of ZFfin that leads to a perspicuous construction of recursive nonstandard !-models of ZFfin without the use of permutations. Furthermore, we show that every recursive model of ZFfin is an !-model. The central theorem of the paper is the following:

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ω-有限集合理论的模型

有限集合论，这里记作ZFfin，是在Zermelo-Fraenkel集合论的通常公化中，用无限公理的否定来代替无限公理而得到的理论。ZFfin的-模型是这样一种模型，其中每个集合最多有有限多个元素(从外部看)。Mancini和Zambella(2001)利用排列的Bernays-Rieger方法构建了一个非标准的ZFfin递归模型(即，与遗传有限集V不同构)。本文对ZFfin的1 -模型进行了元数学研究。特别地，我们提出了一种建立ZFfin -模型的新方法，该方法可以在不使用置换的情况下清晰地构建ZFfin的递归非标准-模型。进一步证明了ZFfin的每一个递归模型都是一个!-模型。本文的中心定理如下:

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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Logic group preprint series

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