Results 1  10
of
11
Forcing in Proof Theory
 BULL SYMB LOGIC
, 2004
"... Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a pla ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbertstyle proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing modeltheoretic arguments.
The epsilon calculus and Herbrand Complexity
 STUDIA LOGICA
, 2006
"... Hilbert’s εcalculus is based on an extension of the language of predicate logic by a termforming operator εx. Two fundamental results about the εcalculus, the first and second epsilon theorem, play a rôle similar to that which the cutelimination theorem plays in sequent calculus. In particular ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Hilbert’s εcalculus is based on an extension of the language of predicate logic by a termforming operator εx. Two fundamental results about the εcalculus, the first and second epsilon theorem, play a rôle similar to that which the cutelimination theorem plays in sequent calculus. In particular, Herbrand’s Theorem is a consequence of the epsilon theorems. The paper investigates the epsilon theorems and the complexity of the elimination procedure underlying their proof, as well as the length of Herbrand disjunctions of existential theorems obtained by this elimination procedure.
Formal semantics of model fields in annotationbased specifications inspired by a generalization of Hilbert’s ɛ terms
, 2011
"... Abstract. It is widely recognized that abstraction and modularization are indispensable for specification of realworld programs. In sourcecode level program specification and verification, model fields are a common means for those goals. However, it remains a challenge to provide a wellfounded for ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
Abstract. It is widely recognized that abstraction and modularization are indispensable for specification of realworld programs. In sourcecode level program specification and verification, model fields are a common means for those goals. However, it remains a challenge to provide a wellfounded formal semantics for the general case in which the abstraction relation defining a model field is nonfunctional. In this paper, we discuss and compare several possibilities for defining model field semantics, and we give a complete formal semantics for the general case. Our analysis and the proposed semantics is based on a generalization of Hilbert’s ε terms. 1
Epsilon substitution for transfinite induction
 Arch. Math. Logic
, 2005
"... We apply Mints ’ technique for proving the termination of the epsilon substitution method via cutelimination to the system of Peano Arithmetic with Transfinite Induction given by Arai. 1 ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
We apply Mints ’ technique for proving the termination of the epsilon substitution method via cutelimination to the system of Peano Arithmetic with Transfinite Induction given by Arai. 1
Simple Structures with Complex Symmetry
, 2009
"... We define the automorphism spectrum of a computable structure M, a measurement of the complexity of the symmetries of M, and prove that certain sets of Turing degrees can be realized as automorphism spectra, while certain others cannot. 1 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
We define the automorphism spectrum of a computable structure M, a measurement of the complexity of the symmetries of M, and prove that certain sets of Turing degrees can be realized as automorphism spectra, while certain others cannot. 1
Proof Theory The Fixed Point Theorem for the Logic of Provability
, 2008
"... I have read and understood the Statement on Plagiarism for Part III and Graduate Courses issued by the Faculty of Mathematics, and have abided by it. This essay is the result of my own work, and except where explicitly stated otherwise, only includes material undertaken since the publication of the ..."
Abstract
 Add to MetaCart
I have read and understood the Statement on Plagiarism for Part III and Graduate Courses issued by the Faculty of Mathematics, and have abided by it. This essay is the result of my own work, and except where explicitly stated otherwise, only includes material undertaken since the publication of the list of essay titles, and includes nothing which was performed in collaboration. No part of this essay has been submitted, or is concurrently being submitted, for any degree, diploma or similar qualification at any university or similar
G.Moser R.Zach The Epsilon Calculus and
, 2005
"... Abstract. Hilbert’s εcalculus is based on an extension of the language of predicate logic by a termforming operator εx. Two fundamental results about the εcalculus, the first and second epsilon theorem, play a rôle similar to that which the cutelimination theorem plays in sequent calculus. In pa ..."
Abstract
 Add to MetaCart
Abstract. Hilbert’s εcalculus is based on an extension of the language of predicate logic by a termforming operator εx. Two fundamental results about the εcalculus, the first and second epsilon theorem, play a rôle similar to that which the cutelimination theorem plays in sequent calculus. In particular, Herbrand’s Theorem is a consequence of the epsilon theorems. The paper investigates the epsilon theorems and the complexity of the elimination procedure underlying their proof, as well as the length of Herbrand disjunctions of existential theorems obtained by this elimination procedure. Keywords: Hilbert’s εcalculus, epsilon theorems, Herbrand’s theorem, proof complexity 1.
An implication of Gödel’s incompleteness theorem
, 2009
"... A proof of Gödel’s incompleteness theorem is given. With this new proof a transfinite extension of Gödel’s theorem is considered. It is shown that if one assumes the set theory ZFC on the meta level as well as on the object level, a contradiction arises. The cause is shown to be the implicit identif ..."
Abstract
 Add to MetaCart
A proof of Gödel’s incompleteness theorem is given. With this new proof a transfinite extension of Gödel’s theorem is considered. It is shown that if one assumes the set theory ZFC on the meta level as well as on the object level, a contradiction arises. The cause is shown to be the implicit identification of the meta level and the object level hidden behind the Gödel numbering. An implication of these considerations is stated.