Lano, K., The intuitionistic alternative set theory, Annals of Pure and Applied Logic 59 141–156 The Alternative Set Theory, as defined in Vopenka and Sochor, demonstrates how a set theory which avoids actually infinite sets can serve as a framework for much of classical mathematics. This paper defines a theory which can serve as an intuitionistic analogue of AST, and examines motivations for alternative formulations of classical AST from an intuitionistic and finitistic viewpoint. The intuitionistic AST uses appropriate modifications of (...) the concepts of AST, with the notion of feasibility replacing finiteness, and with new distinctions between alternative definitions of Countable Class and Revealment. Results of classical AST which are still valid in this new system are given, and an interpretation of the corresponding classical system in the intuitionistic system is defined. This shows that the adoption of an intuitionistic logic does not essentially deprive us of classical methods or results for AST. (shrink)

The connection between similarity logic and the theory of closure operators is examined. Indeed one proves that the consequence relation defined in [14] can be obtained by composing two closure operators and that the resulting operator is still a closure operator. Also, we extend any similarity into a similarity which is compatible with the logical equivalence, and we prove that this gives the same consequence relation.