63 Years of the MacDowell-Specker Theorem

TL;DR

This paper surveys the extensive developments and open problems related to the MacDowell-Specker theorem, which concerns elementary extensions of models of Peano Arithmetic and has influenced many subsequent results.

Contribution

It provides a comprehensive overview of 63 years of research, generalizations, and open questions stemming from the original theorem.

Findings

Historical overview of the theorem's development

Summary of key generalizations and results

Identification of open problems in the area

Abstract

In September of 1959, at the conference on Infinitistic Methods in Warsaw, Ernst Specker presented a joint paper with Robert MacDowell in which the authors proved that every model of Peano Arithmetic has an elementary extension such that all new elements are larger than all elements of the model. Until now, the theorem has been a constant source of new results and generalizations, and there are still open problems related to it. This paper is a survey of all those developments.