A Connection between Leibniz’ Infinitely Small Quantities and the Analytical Hierarchy

Abstract

In this paper I want to show among other things one unexpected connection between the usage of infinitely small quantities and the analytical hierarchy of sets. The usage of infinitely small quantities is due to Leibniz but I use the modern version of the usage by working in a theory similar to the Nelson’s Internal set theory¹. The theory is a weaker one than Nelson’s theory and I think that it contains the minimum for reasonable work with infinitely small quantities. Hence the results can be applied for a large amount of structures and in my opinion it is more similar to the original usage of infinitesimals than e.g. Nelson’s theory just mentioned. A part of the technique which I use was developed in alternative set theory² and hence I mention also some ideas concerning this theory. This is also, in my opinion, in accordance with the main themes of the conference. Using our theory we find an exact, even algorithmical connection of the notions defined with the help of the infinitely small quantities and the corresponding notions definable in the Cantor’s set theory. This theory deals with the actual infinity and it is used as a basis for modern mathematics. In spite of the algorithmical method of exact translations, the translations of the notions defined with the help of infinitely small quantities are comprehensible only in the case of the classical analytical notions such as the limit, continuity, limit point etc. For the more complicated notions (some of them will be specified later) we obtain by known algorithms only incomprehensible notions and I give an argument why this situation takes place. It is interesting that in the translation algorithm introduced by me the complexity of the nonstandard definition has its counterpart in the complexity of the standard translation. The complexity concept mentioned concerns the analytical hierarchy of sets and this is the connection I want to talk about.