Henkin-Keisler Models

Henkin-Keisler models emanate from a modification of the Henkin construction introduced by Keisler to motivate the definition of ultraproducts. Keisler modified the Henkin construction at that point at which "new" individual constants are introduced and did so in a way that illuminates a connection between Henkin-Keisler models and ultraproducts. The resulting construction can be viewed both as a specialization of the Henkin construction and as an alternative to the ultraproduct construction. These aspects of the Henkin-Keisler construction are utilized here to present a perspective on ultraproducts and their applications accessible to the reader familiar with Henkin's proof of the completeness of first order logic and naive set theory. This approach culminates in proofs of various forms of the Keisler-Shelah characterizations of elementary equivalence and elementary classes via Henkin-Keisler models. The presentation is self-contained and proofs of more advanced results from set theory are introduced as needed.