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.