Efficient Metamathematics
Rineke Verbrugge
Abstract:
This thesis contains a number of results in the field of the
metamathematics of first order arithmetic. The focus is on the study
of provability of bounded arithmetic and an alternative definition
of interpretability.
First, Verbrugge introduces notions of efficiency, complexity theory,
bounded arithmetic, provability logic, and interpretability logic, and
some results from the literature necessary in the sequel.
In chapter 3, a proof is presented that under the complexity theoretic
assumption $NP \not = co-NP$ bounded arithmetic does not prove
completeness for all formulas for witness comparison.
In Peano Arithmetic, provable completeness for such formulas plays an
important role in proofs of the formalized version of Rossers Theorem
and Solovay's Completeness Theorem. Verbrugge proves a reflection principle
for ``small'' proves, to derive Rossers Theorem in bounded Arithmetic.
An application of this principle is the Theorem of Bernardi and Montagna.
Chapter 4 returns to the problem of provable completeness
$P \not = NP\cap co-NP$ implicates that Buss' bounded arithmetic
$S^0_1$ does not prove completeness for all $\Sigma^0_1$-sentences.
Chapter 5 presents partial answers to the question:
what is the provability logic of bounded arithmetic?
A class of Kripke frames is used to adapt Solovay's method, and to present
an embedding of models on such frames in bounded arithmetic.
Furthermore Verbrugge proves that the provability logic of bounded
arithmetic is not the modal theory of classical Kripke trees.
In chapter 6 Verbrugge defines {\em feasible interpretability}: the
length of proofs of interpreted axioms is limited by the polynomial
in the length of these axioms. She shows that a number of
interpretations, such as that of $ZF+ {\bf V}={\bf L}$ in $ZF$, are feasible.
On the other hand, not all interpretations can be replaced by feasible
interpretations.
Furthermore, a proof is presented that the interpretability logic $ILM$ is
arithmetically correct and complette for feasible interpretability for
Peano Arithmetic.
Chapter 7 deals with the definitional complexity of feasible
interpretability in Peano Arithmetic. Verbrugge proves with a
combination of a recursion-theoretical reduction and a version of
Lindstr\"om's method with partial truth value definitions, that
feasible interpretability over Peano Arithmetic is $\Sigma^0_2$-complete.