Chapters on Bounded Arithmetic and on Provability Logic Domenico Zambella Abstract: This thesis consists of two parts. The first part is concerned with bounded arithmetic. The first chapter introduces and motivates the research in this dissertation. Extensions of weak fragments of Peano arithmetic to second-order theories are studied. Second-order variables represent finite sets of natural numbers. The investigations are restricted to weak fragments of Peano arithmetic, e.g. theories that cannot prove that the exponentiation function is total. This entails that there are finite sets that cannot be coded with the use of natural numbers, although they can be defined with bounded formulas. A hierarchy of bounded formulas is defined for the number of alternations of second-order bounded quantors. Then, a hierarchy of theories is defined by the introduction of comprehension axioms for formulas in these classes. It is not known whether this hierarchy of bounded formulas is a real hierarchy, i.e. whether it collapses even if we limit ourselves to the standard model. This turns out to be a hard problem, because it is equivalent with the question whether the polynomial hierarchy collapses. A related question is whether the hierarchy of fragments of bounded arithmetic collapses. Although this second problem looks like the first, the relation between them is not fully understood. It is shown that if bounded arithmetic coincides with one of its fragments, then it is provable, in bounded arithmetic, that the polynomial time hierarchy collapses. In the second chapter a fragment of bounded arithmetic of a different kind it discussed. The comprehension axiom is assumed, but the multiplication function is left out. The theory is called linearly bounded arithmetic, because the terms of the language are linear. It is proved that each model of linear arithmetic has an endextension satisfying a fragment of bounded arithmetic in which multiplication is total. However, comprehension is lost. The second part deals with provability logic. In a short introduction, the basic concepts of this area are discussed. Chapter 3 presents new proofs of the arithmetic completeness of \$ILP\$ and \$ILM\$. Albert Visser proved that \$ILP\$ is the modal logic for the interpretability over finitely axiomatisable theories. Volodya Shavrukov and Alessandro Berarducci have shown independently that \$ILM\$ is the interpretability logic for essentially reflexive theories. The prove of these theorems shows the common aspect in them. Chapter 4 is concerned with diagonalizable algebras, in particular with subalgebras of the diagonalizable algebras of arithmetic theories. On the basis of a theorem of Volodya Shavrukov we show that his results can be addapted to prove this theorem for weaker theories as well.