Continuity axiom in Bolzano’s memoir Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwei Werthen, die ein entgegengesetztes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege

Marlena Fila

Abstract


Bernard Bolzano’s paper Rein analytischer Beweis des Lehrsatzes, dass zwischen jezwei Werthen, die ein entgegengesetztes Resultat gewähren, wenigstens eine reelleWurzel der Gleichung liege was published in 1817. It presents a ”purely analytic proof” of the intermediate value theorem for the polynomials of variable x. Aside from polynomials, Bolzano considers other kinds of functions, however the domain of these functions is not clearly defined. In this article, we show that the variable x ranges over real numbers. Specifically, we identify the axioms for the ordered field that Bolzano implicitly applies. We also identify the versions of continuity axiom, and show that while some of them are implicitly applied, others are explicitly stated as ”basics truths”.

Keywords


Bernard Bolzano, Rein analytischer Beweis, continuity axiom, Archimedean axiom

References


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