Results 1 
3 of
3
An Application of Constructive Completeness.
 In Proceedings of the Workshop TYPES '95
, 1995
"... this paper, we explore one possible effective version of this theorem, that uses topological models in a pointfree setting, following Sambin [11]. The truthvalues, instead of being simply booleans, can be arbitrary open of a given topological space. There are two advantages with considering this m ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
this paper, we explore one possible effective version of this theorem, that uses topological models in a pointfree setting, following Sambin [11]. The truthvalues, instead of being simply booleans, can be arbitrary open of a given topological space. There are two advantages with considering this more abstract notion of model. The first is that, by using formal topology, we get a remarkably simple completeness proof; it seems indeed simpler than the usual classical completeness proof. The second is that this completeness proof is now constructive and elementary. In particular, it does not use any impredicativity and can be formalized in intuitionistic type theory; this is of importance for us, since we want to develop model theory in a computer system for type theory. Formal topology has been developed in the type theory implementation ALF [1] by Cederquist [2] and the completeness proof we use has been checked in ALF by Persson [9]. In view of the extreme simplicity of this proof, it might be feared that it has no interesting applications. We show that this is not the case by analysing a conservativity theorem due to Dragalin [4] concerning a nonstandard extension of Heyting arithmetic. We can transpose directly the usual model theoretic conservativity argument, that we sketched above, in this framework. It seems likely that a direct syntactical proof of this result would have to be more involved. The first part of this paper presents a definition of topological models, Sambin's completeness proof, and an alternative completeness proof; we also discuss how Beth models relate to our approach. The second part shows how to use this in order to give a proof of Dragalin's conservativity result; our proof is different from his and, we believe, simpler. In [8] a stronger ...
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 ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
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 ...