Basic Proof Theory, 1st Edition

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Basic Proof Theory, 1st Edition
(Cambridge Tracts in Theoretical Computer Science (Book 43))
A. S. Troelstra, H. Schwichtenberg | 1996 | ISBN: 0521572231 | English | 355 pages | DJVU | 6 MB

The discovery of the set-theoretic paradoxes around the turn of the century, and the resulting uncertainties and doubts concerning the use of high-level abstractions among mathematicians, led D. Hilbert to the formulation of his programme: to prove the consistency of axiomatizations of the essential parts of mathematics by methods which might be considered as evident and reliable because of their elementary combinatorial ("finitistic") character.
Although, by Godel's incompleteness results, Hilbert's programme could not be carried out as originally envisaged, for a long time variations of Hilbert's programme have been the driving force behind the development of proof theory.
The present text attempts to fill a lacuna in the literature. To do so it concentrates on the structural proof theory of first-order logic and its applications, and compares different styles of formalization at some length. A glimpse of the proof theory of first-order arithmetic and second-order logic is also provided, illustrating techniques in relatively simple situations which are applied elsewhere to far more complex systems.
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