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19
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 ..."
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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 ..."
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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 ..."
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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
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 ..."
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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
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 ..."
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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
On Kinds of Indiscernibility in Logic and Metaphysics
, 2011
"... Forthcoming, after an Appendectomy, in the British Journal for the Philosophy of Science. Using the HilbertBernays account as a springboard, we first define four ways in which two objects can be discerned from one another, using the nonlogical vocabulary of the language concerned. (These definit ..."
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Forthcoming, after an Appendectomy, in the British Journal for the Philosophy of Science. Using the HilbertBernays account as a springboard, we first define four ways in which two objects can be discerned from one another, using the nonlogical vocabulary of the language concerned. (These definitions are based on definitions made by Quine and Saunders.) Because of our use of the HilbertBernays account, these definitions are in terms of the syntax of the language. But we also relate our definitions to the idea of permutations on the domain of quantification, and their being symmetries. These relations turn out to be subtle—some natural conjectures about them are false. We will see in particular that the idea of symmetry meshes with a species of indiscernibility that we will call ‘absolute indiscernibility’. We then report all the logical implications between our four kinds of discernibility. We use these four kinds as a resource for stating four metaphysical theses about identity. Three of these theses articulate two traditional philosophical themes: viz. the principle of the identity of indiscernibles (which will come in two versions), and haecceitism. The fourth is recent. Its most notable feature is that it makes diversity (i.e. nonidentity) weaker than what we will call individuality (being an individual): two objects can be distinct but not individuals. For this reason, it has been advocated both for quantum particles and for spacetime points. Finally, we locate this fourth metaphysical thesis in a broader position, which we call structuralism. We conclude with a discussion of the semantics suitable for a structuralist, with particular reference to physical theories as well as elementary model theory.
My fourty years on his shoulders
, 2007
"... Gödel's legacy is still very much in evidence. We will not attempt to properly discuss the full impact of his work and all of the ongoing important research programs that it suggests. This would require a book length manuscript. Indeed, there are several books discussing the Gödel legacy from m ..."
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Gödel's legacy is still very much in evidence. We will not attempt to properly discuss the full impact of his work and all of the ongoing important research programs that it suggests. This would require a book length manuscript. Indeed, there are several books discussing the Gödel legacy from many points of view, including, for example, [Wa87], [Wa96], [Da05], and the historically comprehensive five volume set [Go,8603]. In sections 27 we briefly discuss some research projects that are suggested by some of his most famous contributions. In sections 811 we discuss some highlights of a main recurrent theme in our own research, which amounts to an expansion of the Gödel incompleteness phenomenon in a critical direction.