Following, and with our modifications, we expose the claimed (but not right) Cantorian proof. At first he proposes the unacceptability of 1/ ω, ω being the first transfinite ordinal. It may therefore seem strange that he does not accept the actual infinitesimals. Without this approach, he could not have constructed this theory. As we know, Cantor conceives the actual infinite which plays a crucial role in his theory of transfinite numbers. We will refer to a letter written by Cantor to Benno Kerry, dated 4 February 1887. 6 2 Cantor and the unacceptability of the infinitesimalsħCantor examines the unacceptability of the actual infinitesimals. Then we highlight the debate on non-Archimedean concepts in Italy.
6 For further information see infra, , [Borga, Freguglia (.)ĦIn this article, first we present the Cantorian attempt to show the unacceptability of the infinitesimal notion.In fact real numbers are based on the Dedekind axiom and it is well known that:Īrchimedes Post. The inverse is an infinity: 1/ (1/ n ) = n| n =∞ = ∞.ģWe will define below the notion of the actual infinitesimal.ĤTo understand the notion of infinitesimal we must take into account the concept of real numbers. E.g., a potential infinitesimal is 1/n |n=∞. Infinitesimals and infinities naturally occur in any description of the infinity. The infinitesimal can be seen as its reciprocal. ĢOnce the infinite number is accepted as an infinity, this number does not stay in R. But they will only believe in the infinitely small with difficulty, despite the fact that the infinitely small has the same right to existence as the infinitely large. A majority of educated people will admit an infinite in space and time, and not just an “unboundedly large”. A belief in the infinitely small does not triumph easily. The infinitely small is a mathematical quantity and has all its properties in common with the finite. On that subject, Paul du Bois-Reymond wrote: A potentially very small magnitude can be generated through the principle according to which given a magnitude there is always a smaller one (see Archimedes’ postulate). 4 Likewise, we can state concerning the infinitesimals. For Aristotle the actual infinity cannot be accepted, and this conception prevailed until the nineteenth century. 3 So, for instance, in Euclidean geometry, a half-straight line is generated by applying the first and second Euclidean postulates. The potential infinite is conceived as something to which it is always possible to add a certain quantity, while the actual infinite is the possibility of instantly imagining a collection, a whole that has no end. At first we think it is useful to outline the distinction between actual infinite and potential infinite, according to the classical tradition. 2 Mathematicians who were interested in the foundations of mathematics often had different opinions. 4 About the actual infinite, Desargues had written: “So every straight line is intended to be stretch (.)ġBetween the 19th and 20th centuries the possibility of theoretically accepting, or not, actual infinitesimals was much debated.