2 research outputs found
Herbrand Consistency of Some Arithmetical Theories
G\"odel's second incompleteness theorem is proved for Herbrand consistency of
some arithmetical theories with bounded induction, by using a technique of
logarithmic shrinking the witnesses of bounded formulas, due to Z. Adamowicz
[Herbrand consistency and bounded arithmetic, \textit{Fundamenta Mathematicae}
171 (2002) 279--292]. In that paper, it was shown that one cannot always shrink
the witness of a bounded formula logarithmically, but in the presence of
Herbrand consistency, for theories with , any witness for any bounded formula can be shortened logarithmically. This
immediately implies the unprovability of Herbrand consistency of a theory
in itself.
In this paper, the above results are generalized for . Also after tailoring the definition of Herbrand
consistency for we prove the corresponding theorems for . Thus the Herbrand version of G\"odel's second incompleteness
theorem follows for the theories and
Bounded Arithmetic in Free Logic
One of the central open questions in bounded arithmetic is whether Buss'
hierarchy of theories of bounded arithmetic collapses or not. In this paper, we
reformulate Buss' theories using free logic and conjecture that such theories
are easier to handle. To show this, we first prove that Buss' theories prove
consistencies of induction-free fragments of our theories whose formulae have
bounded complexity. Next, we prove that although our theories are based on an
apparently weaker logic, we can interpret theories in Buss' hierarchy by our
theories using a simple translation. Finally, we investigate finitistic G\"odel
sentences in our systems in the hope of proving that a theory in a lower level
of Buss' hierarchy cannot prove consistency of induction-free fragments of our
theories whose formulae have higher complexity