Abstract
This chapter discusses the infinitary methods in the model theory of set theory and also presents the study of the end extensions of models of Zermelo-Fraenkel (ZF) set theory. The theorem of Keisler-Morley, which states that every countable model of ZF has a proper elementary end extension, is focussed. It is shown that if ZF is consistent then there are uncountable models of ZF with no end extensions. The necessary preliminaries are described. All the results are proved using methods and results from infinitary logic. Some of the result on collapsing cardinals used the compactness theorem to prove the theorem of Friedman.
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