A Tableau-Based Decision Procedure for a Fragment of Set Theory with Iterated Membership

2005 ◽  
Vol 34 (1) ◽  
pp. 49-72 ◽  
Author(s):  
Domenico Cantone ◽  
Calogero G. Zarba ◽  
Rosa Ruggeri Cannata
Keyword(s):  
Set Theory ◽  
Decision Procedure

scholarly journals The interpretability logic of Peano arithmetic

1990 ◽  
Vol 55 (3) ◽  
pp. 1059-1089 ◽  
Author(s):  
Alessandro Berarducci
Keyword(s):  
Modal Logic ◽  
Modal Analysis ◽  
Set Theory ◽  
Decision Procedure ◽  
Peano Arithmetic ◽  
Relevant Modal Logic ◽  
Binary Operator ◽  
Base Theory ◽  
Interpretability Logic ◽  
Work Done

AbstractPA is Peano arithmetic. The formula InterpPA(α, β) is a formalization of the assertion that the theory PA + α interprets the theory PA + β (the variables α and β are intended to range over codes of sentences of PA). We extend Solovay's modal analysis of the formalized provability predicate of PA, PrPA(x), to the case of the formalized interpretability relation InterpPA(x, y). The relevant modal logic, in addition to the usual provability operator ‘□’, has a binary operator ‘⊳’ to be interpreted as the formalized interpretability relation. We give an axiomatization and a decision procedure for the class of those modal formulas that express valid interpretability principles (for every assignment of the atomic modal formulas to sentences of PA). Our results continue to hold if we replace the base theory PA with Zermelo-Fraenkel set theory, but not with Gödel-Bernays set theory. This sensitivity to the base theory shows that the language is quite expressive. Our proof uses in an essential way earlier work done by A. Visser, D. de Jongh, and F. Veltman on this problem.

A Course on Set Theory

2009 ◽  
Author(s):  
Ernest Schimmerling
Keyword(s):  
Set Theory

Set Theory

2016 ◽  
Author(s):  
Daniel W. Cunningham
Keyword(s):  
Set Theory
Sign in / Sign up

Export Citation Format

Share Document