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Any intermediate propositional logic (i.e., a logic including intuitionistic logic and contained in classical logic) can be extended to a calculus with epsilon- and tau-operators and critical formulas. For classical logic, this results in Hilbert’s
$\varepsilon $
-calculus. The first and second
$\varepsilon $
-theorems for classical logic establish conservativity of the
$\varepsilon $
-calculus over its classical base logic. It is well known that the second
$\varepsilon $
-theorem fails for the intuitionistic
$\varepsilon $
-calculus, as prenexation is impossible. The paper investigates the effect of adding critical
$\varepsilon $
- and
$\tau $
-formulas and using the translation of quantifiers into
$\varepsilon $
- and
$\tau $
-terms to intermediate logics. It is shown that conservativity over the propositional base logic also holds for such intermediate
${\varepsilon \tau }$
-calculi. The “extended” first
$\varepsilon $
-theorem holds if the base logic is finite-valued Gödel–Dummett logic, and fails otherwise, but holds for certain provable formulas in infinite-valued Gödel logic. The second
$\varepsilon $
-theorem also holds for finite-valued first-order Gödel logics. The methods used to prove the extended first
$\varepsilon $
-theorem for infinite-valued Gödel logic suggest applications to theories of arithmetic.
We give examples of calculi that extend Gentzen’s sequent calculus LK by unsound quantifier inferences in such a way that (i) derivations lead only to true sequents, and (ii) proofs therein are nonelementarily shorter than LK-proofs.
We consider the following problem: Given a proof of the Skolemization of a formula F, what is the length of the shortest proof of F? For the restriction of this question to cut-free proofs we prove corresponding exponential upper and lower bounds.
Kurt Gödel and the Foundations of Mathematics: Horizons of Truth is the culmination of a creative research initiative coorganized by the Kurt Gödel Society, Vienna; the Institute for Experimental Physics; the Kurt Gödel Research Center; the Institute Vienna Circle; the Vienna University of Technology; the Austrian Academy of Sciences; and the Anton Zeilinger Group at the University of Vienna, where the Gödel centenary celebratory symposium “Horizons of Truth: Logics, Foundations of Mathematics, and the Quest for Understanding the Nature of Knowledge” was held from April 27 to April 29, 2006.
More than twenty invited world-renowned researchers in the fields of mathematics, logic, computer science, physics, philosophy, theology, and the history of science attended the symposium, giving the participants the remarkable opportunity to present their ideas about Gödel's work and its influence on various areas of intellectual endeavor. These fascinating interdisciplinary lectures provided new insights into Gödel's life and work and their implications for future generations of researchers.
The interaction among international scholars who only rarely, if ever, have the opportunity to hold discussions in the same room – and some of whom almost never write articles – has produced a book that contains chapters expanded and developed to take advantage of the rich intellectual exchange that took place in Vienna. Written by some of the most renowned figures of the scientific and academic world, the resulting volume is an opus of current research and thinking that is built on the work and inspiration of Gödel.