Internal Set Theory IST# Based on Hyper Infinitary Logic with Restricted Modus Ponens Rule: Nonconservative Extension of the Model Theoretical NSA
Jaykov Foukzon *
Israel Institute of Technology, Haifa, Israel.
*Author to whom correspondence should be addressed.
Abstract
The incompleteness of set theory \(Z F C\) leads one to look for natural nonconservative extensions of \(Z F C\) in which one can prove statements independent of \(Z F C\) which appear to be "true". One approach has been to add large cardinal axioms.Or, one can investigate second-order expansions like Kelley-Morse class theory, \(K M\) or Tarski-Grothendieck set theory \(T G\) or It is a nonconservative extension of \(Z F C\) and is obtained from other axiomatic set theories by the inclusion of Tarski's axiom which implies the existence of inaccessible cardinals. See also related set theory with a filter quantifier \(Z F(a a)\). In this paper we look at a set theory \(\mathrm{NC}_{\infty}^{\#}\), based on bivalent gyper infinitary logic with restricted Modus Ponens Rule In this paper we deal with set theory \(\mathrm{NC}_{\infty}^{\#}\) based on bivalent gyper infinitary logic with Restricted Modus Ponens Rule. Nonconservative extensions of the canonical internal set theories IST and HST are proposed.
Keywords: Set theory ZFC, nonconservative extension of ZFC, internal set theory IST, external settheory HST, A. Robinson model theoretical NSA, Bivalent gyper in nitary logic, odusponens rule, Logic with restricted, modus ponens rule