Title:
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Some uses of cut elimination
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This thesis is mainly about Proof Theory. It can be thought of as Proof
Theory in the sense of Hilbert, Gentzen, Schutte, Buchholz, Rathjen, and
in general what could be called the German school, but it is also influenced
by many other branches, of which the bibliography might give an idea.
Intuitionism and other philosophical approaches to mathematics are also an
important part of what is studied, but the Leitmotif of this thesis is Cut
Elimination. The first part of the thesis is concerned with countable coded
ω-models of Bar Induction. In this part we work from a reverse mathematics
point of view. A study for an ordinal analysis of the theory of Bar Induction
(BI) is carried out, and the equivalence between the statement that every
set is contained in an co-model of this theory (BI) and the well-ordering
principle VX[W0(3E) WO(6x)] which says that if X is a well-ordering,
then so is its Bachmann-Howard relativisation, is proven. This is a new
result as far as we know, and, we hope, an important one. In the second
part of the thesis we shift our viewpoint and consider intuitionistic logic
and intuitionistic geometric theories. We show that geometric derivability in
classical infinitary logic implies derivability in intuitionistic infinitary logic.
Again, our main tool is Cut Elimination. Next, we present investigations
regarding minimal logic and classical logical principles, and give a precise
classification of excluded middle, ex falso, and double negation elimination.
Other themes and roads are possible and, the author feels, important, but
time limitations as well as a sickly and utterly daft adherence to deadlines
did not permit him to carry out these studies in full. It is quite shameful.
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