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A New Approach to Predicative Set Theory
"... We suggest a new basic framework for the WeylFeferman predicativist program by constructing a formal predicative set theory PZF which resembles ZF. The basic idea is that the predicatively acceptable instances of the comprehension schema are those which determine the collections they define in an a ..."
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We suggest a new basic framework for the WeylFeferman predicativist program by constructing a formal predicative set theory PZF which resembles ZF. The basic idea is that the predicatively acceptable instances of the comprehension schema are those which determine the collections they define in an absolute way, independent of the extension of the “surrounding universe”. This idea is implemented using syntactic safety relations between formulas and sets of variables. These safety relations generalize both the notion of domainindependence from database theory, and Godel notion of absoluteness from set theory. The language of PZF is typefree, and it reflects real mathematical practice in making an extensive use of statically defined abstract set terms. Another important feature of PZF is that its underlying logic is ancestral logic (i.e. the extension of FOL with a transitive closure operation). 1
Local Universes in the system G
"... This paper is concerned with the theory of local universes, where universal properties of the universe of sets are restricted to local sets satisfying relevant closure properties. The notion of local universe is derived from the wellknown theory of admissible sets, that it is usually presented in t ..."
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This paper is concerned with the theory of local universes, where universal properties of the universe of sets are restricted to local sets satisfying relevant closure properties. The notion of local universe is derived from the wellknown theory of admissible sets, that it is usually presented in terms of models of special subtheories of sets (see [1] and [3]). We have found that in the context of the theory G (see [6]), where the universe is actually rejected as a complete totality, it is more natural to think of admissible sets as local structures (sets) that have some of the closure properties of the universe. This may appear to be a shameless attemp to eat the cake (rejecting the universe) and to keep it (via pseudo universes). In reality, we reject the universe only when it is involved in the actual definition of sets or predicates in the universe. We consider the universe to be an ambigious totality that the theory explores without ever reaching its limits.
Completeness in Operational Set Theory
"... Introduction This note is concerned with operational set theory, a concept that was introduced in Sanchis [8], via the sysstem G, a partially formalized set theory. The leit motiv of this approach is the predicativity principle, that we understand as a general principle in the sense that the univers ..."
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Introduction This note is concerned with operational set theory, a concept that was introduced in Sanchis [8], via the sysstem G, a partially formalized set theory. The leit motiv of this approach is the predicativity principle, that we understand as a general principle in the sense that the universe of sets is not a well defined totality, and quantification over the universe may be allowed only under severe logical and constructive restrictions. This approach to set theory implies a number of constraints that are enforced in [8]. Here we concentrate our attention only on some crucial consequences, namely the peculiar axiomatic structure that it is induced by this approach. We say that an axiom in set theory is operational if it is a local formula involving local quantifiers but not global quantifiers. Furthermore, we consider that the universal closure of an operational axiom is also an operational axiom. An opera
Primitive Constructions in Set Theory
, 1999
"... this paper we propose a general theory of primitive constructions involving reductions (see Section 3), and review the constructions in [4] in the context of this theory. We are particularly 2 ..."
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this paper we propose a general theory of primitive constructions involving reductions (see Section 3), and review the constructions in [4] in the context of this theory. We are particularly 2
Logical Foundations for Operational Set Theory
"... this paper we take the position that the universe of sets is a fiction that it is not necessary in order to explain the global quantifiers, provided we restrict their use via the logical structure of the system. At the same time the local quantifiers become the real protagonists of set theory, which ..."
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this paper we take the position that the universe of sets is a fiction that it is not necessary in order to explain the global quantifiers, provided we restrict their use via the logical structure of the system. At the same time the local quantifiers become the real protagonists of set theory, which is now a theory of sets and not a theory of a mythical universe of sets