Vopěnka’s Alternative Set Theory within Twentieth Century Mathematics

Abstrakt

Petr Vopěnka presented his Alternative Set Theory (AST) in the monograph, Mathematics in the Alternative Set Theory, published by Teubner Verlag in 1979. Prior to this publication, the AST had been developed for a decade in a research seminar headed by himself and attended by a group of graduate students and collaborators, including Antonín Sochor, Josef Mlček, or Karel Čuda, and these explorations yielded dozens of papers available via the Czech Digital Mathematical Library (dml.cz). As a sidenote, during this decade Vopěnka was prevented from publishing his own work, maintaining international scientific contacts, or developing his career, for political reasons. Vopěnka subsequently published another monograph on the AST in 1989, and offered several papers and lectures that detailed the philosophical and historical motivations for the AST. The latter include the works of Leibniz and Bolzano; as a matter of fact, Vopěnka’s interests in various areas of the history of mathematics were quite broad, producing works ranging from geometry to analysis and set theory. However, the AST also has tight links to developments in mathematics in the 20th century, roughly from the 1930’s onwards: namely, Skolem’s work in nonstandard models of arithmetic, Vopěnka’s nonstandard models of set theory, Robinson’s nonstandard analysis, Vopěnka and Hájek’s work in the theory of semisets, or Parikh’s feasible arithmetic. These are only some major mathematical influences; the AST also reflects Vopěnka’s growing interest in phenomenology. Moreover, one can find parallel developments that were presumably independent of the AST but share some of its motivations, background, and its timing, such as the axiomatic nonstandard set theories proposed by Nelson and by Hrbáček. This talk will present and discuss the twentieth century mathematical narrative for the AST and demonstrate how the AST can be understood as a natural continuation of the above inspirations and experience.