Abstract: This chapter introduces some techniques for constructing elementary simple extensions. In particular, it includes proof of the existence of superminimal elementary end extensions of countable models and a construction of a Jónsson model. The MacDowell-Specker Theorem is proved and some of its consequences derived, including results on conservative extensions, rather classless models and amalgamations. The chapter concludes with a discussion of nonelementary extensions and the theorem of Wilkie on end extensions which are not ₁-elementary.

Keywords: simple extensions, elementary extensions, minimal extensions, MacDowell-Specker Theorem, conservative extensions, amalgamations,