The iterative conception of set commonly is regarded as supporting the axioms of Zermelo-Fraenkel set theory (ZF). This paper presents a modified version of the iterative conception of set and explores the consequences of that modified version for set theory. The modified conception maintains most of the features of the iterative conception of set, but allows for some non-wellfounded sets. It is suggested that this modified iterative conception of set supports the axioms of Quine's set theory NF.

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)

Provides an overview of Mark Sharlow's philosophical work with summaries of his positions. Includes references and links to his writings.