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Vopenka [3]. First of all there are semisets (proper subclasses of sets) here and a theorem can be proved that a class is standard if and only if its intersection with any standard set is a standard set. It is an open question whether a similar theorem for internal classes is true as well. Various other notions which stem from the concept of natural infinity can be formalized in NCT, but in contrast with AST it preserves all results of the classical mathematics. Thus, in a way NCT can be viewed as an alternative to the Alternatice Set Theory.

CFIS, 626 Rhodes Hall, Cornell University. Ithaca, NY 14853-3801. USA. E-mail: davorenahybrid . edu. An abstract transition system is a structure I = (X, { $ ) i , t ~ c t ) . where X # 0 is the state space and for each a E Act. 5 is a binary relation on X . Such structures are used in many areas of computer science to model dynamic changes of state. They provide a Kripke semantics for propositional modal languages with box modalities [a] for a E Act. Given an assignment of a subset of X to each atomic proposition, extend the valuation to all formulas by interpreting the propositional connectives by their set-theoretic counterparts.

They enclose both commutative and non commutative multiplicative connectives: Z (Then)is the non commutative tensor. (Sequential)is the non commutative counterpart of Par, and ':' is the non commutative ','. Such systems have been introduced by Philippe de Groote [2] (limited to the multiplicative intuitionistic fragment) and Paul Ruet [3](who exhaustively studies the full classical system, and its intuitionistic pendant). Yet the latter system has properties which we believe are nondesirable: commutativity between serialized formulas is enabled by the exponentials !

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1998 European Summer Meeting of the Association for Symbolic Logic. Logic Colloquium '98

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