Completeness of Propositional Provability Logic

Propositional provability logic stems from the desire to investigate what mathematical theories can say about themselves. In this thesis, we will discuss multiple papers on the completeness of propositional provability logic. It turns out that propositional provability logic is complete with respect to the class of finite irreflexive trees and arithmetically complete with respect to Peano Arithmetic. The last result gave rise to exploring the boundaries of propositional provability logic with respect to weaker arithmetics. We will discuss a class of theories discovered by D. de Jongh who has proven its arithmetical completeness based on the proof that R.M. Solovay provided for Peano Arithmetic.