Models of Set Theory: Extensions and Dead-ends

TL;DR

This paper explores extensions of models of ZF set theory, introducing new constructions and demonstrating the existence of models that cannot be further extended to ZF models.

Contribution

It provides novel methods for constructing extensions of ZF models and proves the existence of models that cannot be extended further.

Findings

New constructions of ZF model extensions

Existence of non-extendable ZF models

Extensions do not require countability or well-foundedness

Abstract

This paper is a contribution to the study of extensions of arbitrary models of ZF (Zermelo-Fraenkel set theory), with no regard to countability or well-foundedness of the models involved. We present some new constructions of certain types of extensions, and also establish the existence of models of ZF that cannot be properly end extended to a model of ZF.