Linear set theory with strict comprehension
Masaru Shirahata · 1998
In this paper, we study the extensionality axiom in the set theory with the unrestricted comprehension based on linear logic. We first review Grishin's result which shows the imcompatibility of the extensionality axiom and the unrestricted comprehesion in linear set theory. As one way to rectify this situation, we introduce the notion of "strict comprehension" and formulate a system of linear set theory which contains the extensionality and the strict comprehesion. The consistency of such a system is then proved by a simple cut-elimination argument. 1 Introduction The set theory based on contraction-free logics [1, 5] has been studied for the reason that the use of unrestricted comprehension does not necessarily yield the contradiction in such logics [2, 4, 6, 7, 9]. However, it has been also noticed that the standard extetensionality axiom cannot be added to such a set theory without causing inconsistency [3, 10]. In this paper, we show one way to remedy this situation, using the not...