On ω-models which are not β-models
Andrzej Włodzimierz Mostowski, Y. Suzuki · Fundamenta Mathematicae · 1969
Publisher Summary This chapter proves a theorem stating that β -models for the second-order arithmetic cannot be distinguished from ω- models by elementary sentences. It determines a similar result for models of the Zermelo–Fraenke1 set theory and gives a solution of a problem concerning the existence of models which are א τ -standard but are not א τ+1 -standard. The chapter presents abbreviations and definitions and formulates a few theorems that can be proved in the basis of defined axioms. It defines semantical notions of satisfaction, model, elementary extension, reduct, and diagram. It describes the pigeon-hole principle, which states that if many objects are put into a small number of drawers, then at least one drawer contains many objects. A new family of non-standard models is constructed for set theory in the chapter.