On the Foundations of a Unified Theory including Set Theory , Non - Standard Analysis and Finite Analysis

The paper shows how the principle that the whole must be greater than the part is not necessarily inconsistent with being bijected with a proper subset, provided that equicardinality is reinterpreted as related with definability and not with sameness of size. An explanation for such reinterpretation is offered on the basis of availability, which leads to the problem of graduality, as raised by the Sorites Paradox, and to Finite Analysis, as developed by S. Lavine from the finiteness theorems of J. Mycielski. In order to have a formal version of availability, non-standard analysis is then considered: first in its classic, model theoretic version, showing that Robinson’s infinities are not Dedekind’s, and later in its axiomatic version, E. Nelson’s Internal Set Theory (IST), which can be used not only to prove the existence of infinitesimals but also, as shown in this paper, to deduce many properties about infinity, proving as a theorem ZFC’s Axiom of Infinity. IST is shown to contain standard “natural” numbers with non-standard predecessors that must be considered (Dedekind) infinite, requiring a redefinition of infinitesimals and their inverses. With such redefinition, IST is compatible with J. H. Conway’s surreal numbers, providing a foundation for a unified number system including the traditional Cantorian transfinite ordinals as well as their infinitesimal inverses. The paper argues why the new number system should be used inside IST–AI, instead of inside ZFC. Such number system is shown, to conclude, as being able to provide a measure for the size of any kind of sets, the cardinals no longer providing such a measure under the proposed reinterpretation.