Pairs, sets and sequences in first order theories

Abstract

In this paper we study the idea of theories with containers, like sets, pairs, sequences. We provide a modest framework to study such theories. We prove two concrete results. First, we show that first order theories of finite signature that have functional non-surjective ordered pairing are definitionally equivalent to extensions in the same language of the basic theory of non-surjective ordered pairing. Secondly, we show that a first order theory of finite signature is sequential (is a theory of sequences) if it is definitionally equivalent to an extension in the same language of a system of weak set theory called WS.