Andrej Ščedrov

The lack of definable witnesses and provably recursive functions in intuitionistic set theories

Abstract Let ZFIR(ZFIC) be intuitionistic ZF set theory formulated with Replacement (resp. Collection). It is known that if ZFIR proves a sentence ∃xA(x), then there is a formula C(z) so that ZFIR proves ∃!zC(z) and ∃x(C(x) ∧ A(x)), the existence property. It is shown that ZFIC does not have the existence property, and thus ZFIR ⫋ ZFIC. This remains true even if one adds Dependent Choice and all true Σ1 sentence of ZF. It is known that ZF and ZFIc have the same provably recursive functions. It is also shown that this is not true for ZFIC and ZFIR.

Intuitionistic Set Theory

Publisher Summary This chapter describes Friedmans contributions to intuitionistic set theory. These contributions include Friedmans extension of Gaels negative interpretation and Friedmans extension of Kleenes recursive realizability. One of the first significant results about intuitionistic systems was obtained in 1932 by Godel who gave a syntactical translation of classical predicate calculus into Heytings predicate calculus. Thus the consistency of a system with classical logic is reduced to the consistency of a system with intuitionistic logic, and furthermore the classical system can be viewed as a subsystem (or a special case) of an intuitionistic one. Finally, the chapter discusses the partially intuitionistic fragments of ZFC for which Excluded Middle holds for an important class of formulas.

Classifying topoi and finite forcing

We show that Robinson’s finite forcing, for a theory +rT, is a universal construction in the sense of categorical algebra: it is the satisfaction relation for the universal model in the classifying topos 6 of a certain universal Horn theory defined from .K Assuming, without loss of generality, that .I- is axiomatized by universal sentences, we construct, as sheaf subtopoi of 6, the classifying topoi for (i.e., universal examples of) finitely generic models, existentially closed models, and arbitrary models of .I- (with complemented primitive predicates).

Set existence property for intuitionistic theories with dependent choice

Abstract Let TC be intuitionistic higher-order arithmetic or intuitionistic ZF (with Replacement), both with Relativized Dependent Choice, or just Countable Choice. We show that TC[boxvr]∄ x . A( x ) (closed) gives TC[boxvr]A( t ) for some comprehension term t . This solves a problem left open by Myhill in [4].

Large sets in intuitionistic set theory

Abstract We consider properties of sets in an intuitionistic setting corresponding to large cardinals in classical set theory. Adding such ‘large set axioms’ to intuitionistic ZF set theory does not violate well-know metamathematical properties of intuitionistic systems. Moreover, we consider statements in constructive analysis equivalent to the consistency of such ‘large set axioms’.

Complete topoi representing models of set theory

Abstract By a model of set theory we mean a Boolean-valued model of Zermelo-Fraenkel set theory allowing atoms (ZFA), which contains a copy of the ordinary universe of (two-valued,pure) sets as a transitive subclass; examples include Scott-Solovay Boolean-valued models and their symmetric submodels, as well as Fraenkel-Mostowski permutation models. Any such model M can be regarded as a topos. A logical subtopos E of M is said to represent M if it is complete and its cumulative hierarchy, as defined by Fourman and Hayashi, coincides with the usual cumulative hierarchy of M. We show that, although M need not be a complete topos, it has a smallest complete representing subtopos, and we describe this subtopos in terms of definability in M. We characterize, again in terms of definability, those models M whose smallest representing topos is a Grothendieck topos. Finally, we discuss the extent to which a model can be reconstructed when its smallest representing topos is given.

Extending Gödel's Modal Interpretation to Type Theory and Set Theory

Publisher Summary Heytings propositional calculus can be embedded in Lewiss modal propositional calculus S4, as exhibited by Godel. Thus, S4 conservatively contains both classical and intuitionistic propositional calculi. It is also observed that using topological Boolean algebras, a similar interpretation can be extended to the predicate calculus without equality. When attempting to build type theory or set theory based on S4, one has to be careful with comprehension. The chapter describes an extensional type theory based on S4. This theory exhibits existence and disjunction properties and the intiutionistic type theory can be interpreted in it. The chapter considers the topological interpretation of S4 logic without restricting to the first-order case. The intuitionistic Zermelo-Fraenkel set theory (ZF) can be interpreted in set theory based on S4. The chapter also formulates a higher-order S4 calculus containing all classical, intuitionistic, and modal higher-order predicates.

On the quantificational logic of intuitionistic set theory

Defaut de maximalite du calcul des precedents de Heyting pour les theories ensemblistes intuitionistes faibles. Distinction logique de collection