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Saturated models of universal theories
 Annals of Pure and Applied Logic
"... A notion called Herbrand saturation is shown to provide the modeltheoretic analogue of a prooftheoretic method, Herbrand analysis, yielding uniform modeltheoretic proofs of a number of important conservation theorems. A constructive, algebraic variation of the method is described, providing yet a ..."
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A notion called Herbrand saturation is shown to provide the modeltheoretic analogue of a prooftheoretic method, Herbrand analysis, yielding uniform modeltheoretic proofs of a number of important conservation theorems. A constructive, algebraic variation of the method is described, providing yet a third approach, which is finitary but retains the semantic flavor of the modeltheoretic version. 1
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 ..."
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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.
Formal Topologies on the Set of FirstOrder Formulae
 Journal of Symbolic Logic
, 1998
"... this paper that the question has a simple negative answer. This raised further natural questions on what can be said about the points of these two topologies; we give some answers. The observation that topological models for firstorder theories can expressed in the framework of locales appears, for ..."
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this paper that the question has a simple negative answer. This raised further natural questions on what can be said about the points of these two topologies; we give some answers. The observation that topological models for firstorder theories can expressed in the framework of locales appears, for instance, in Fourman and Grayson [6], where the analogy between points of a locale and models of a theory is emphasised; the identification of formal points with Henkin sets, gives a precise form to this analogy. We replace the use of locales by formal topology, which can be expressed in a predicative framework such as MartinLof's type theory. Prooftheoretic issues are also considered by Dragalin [4], who presents a topological completeness proof using only finitary inductive definitions. Palmgren and Moerdijk [10] is also concerned with constructions of models: using sheaf semantics, they obtain a stronger conservativity result than the one in [3]. We will first investigate the difference between the DedekindMacNeille cover and the inductive cover. It easy to see that \Delta DM is stronger than \Delta I , that is, OE \Delta I U implies OE \Delta DM U , but the converse does not hold in general. The notion of point is not primitive in formal topology and therefore it is natural to require that a formal topology has some notion of positivity defined on the basic neighbourhoods; that a neighbourhood is positive then corresponds to, in ordinary point based topology, that it is inhabited by some point. We will show several negative results on positivity, both for the inductive topology and the DedekindMacNeille topology. The points of an inductive topology correspond to Henkin sets, but the DedekindMacNeille topology has, in general, no points. Our reasoning is constructi...
An Effective Conservation Result for Nonstandard Arithmetic
, 1999
"... We prove that a nonstandard extension of arithmetic is effectively conservative over Peano arithmetic by using an internal version of a definable ultrapower. By the same method we show that a certain extension of the nonstandard theory with a saturation principle has the same prooftheoretic strengt ..."
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We prove that a nonstandard extension of arithmetic is effectively conservative over Peano arithmetic by using an internal version of a definable ultrapower. By the same method we show that a certain extension of the nonstandard theory with a saturation principle has the same prooftheoretic strength as second order arithmetic, where comprehension is restricted to arithmetical formulas. Mathematics Subject Classification: 03F30, 03H15. Keywords: Nonstandard arithmetic, prooftheoretic strength, bounded ultrapowers. 1 Introduction While it is trivially true that nonstandard methods used inside ZFC will not increase the set of theorems that ZFC proves, the addition of axioms corresponding to transfer and saturation principles to weaker theories may, or may not, increase the strength of the theory. A conservation result for a higher order theory weaker than ZFC was obtained by Kreisel [7]. The prooftheoretic strength of various axioms for nonstandard arithmetic, especially saturation ...