By Michael Hallett

Cantor's rules shaped the foundation for set concept and likewise for the mathematical therapy of the concept that of infinity. The philosophical and heuristic framework he constructed had an enduring influence on glossy arithmetic, and is the recurrent subject matter of this quantity. Hallett explores Cantor's rules and, particularly, their ramifications for Zermelo-Frankel set conception.

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Self-conjugacy of an operator is also equivalent to the equation x · f −f x = 0, corresponding to the Brouwerian modal axiom p → ✷✸p. In closure algebras this is equivalent to every closed element being open: a self-conjugate closure algebra is the same thing as a monadic algebra. As already mentioned, this study of BAO’s was later overlooked. e. antisymmetric quasi-orderings) and closure operations of certain topologies on a set. 1). 4 Could Tarski Have Invented Kripke Semantics? A question like this can only remain a matter of speculation.

F. William Lawvere’s search for categorical axioms for set theory and the foundations of mathematics and his collaboration with Miles Tierney on axiomatic sheaf theory culminated at the end of the decade in the development of elementary topos theory. g. [Montague, 1970, fn. 5]. Mathematical Modal Logic: A View of its Evolution 37 of Kripke, Cohen, Scott, and Solovay, as well as incorporating the sheaf theory of the Grothendieck school of algebraic geometry. Scott’s construction of models for the untyped lambda calculus in 1969 was to open up the discipline of denotational semantics for programming languages, as well as stimulating new investigations in lattice theory and topology, and further links with categorical and intuitionistic logic.

Scott then made this material available in a mimeographed form [Lemmon and Scott, 1966] which was circulated informally for a number of years, becoming known as the “Lemmon Notes”. Eventually it was edited by Scott’s student Krister Segerberg, and published as [Lemmon, 1977]. ) in discussion with Montague, Kaplan and others. Some of his ideas were presented in [Scott, 1970]. His considerable inﬂuence on the subject has been disseminated through the publications of Lemmon and Segerberg, and is also reported in [Prior, 1967] in relation to tense logic, and in a number of Montague’s papers.