Aritmetica di Frege. Parte I: Teoria delle serie

[Autom. eng. transl.] The present study constitutes the first part of a larger project dedicated to an analysis, carried out under the logical and philosophical profile, of Frege's arithmetic. Frege's arithmetic is a theory developed in the context of second-order logic and defined by the introduction of a single principle, known in the literature as the principle of Hume, which states that the numbers associated with two concepts are equal if and only if the two concepts can be placed in one-to-one correspondence. The text is divided into four chapters, in which are presented: 1) the logic system; 2) the elementary theory of relations and correspondence relations; 3) the fundamental theorems with reference to the closing relations and to the series; 4) the correspondence theorems between infinite series and finite series. The main contributions to the current discussion are of three types: first, explicit and complete proofs of the indicated theorems are presented, that is proofs that, according to the requirement highlighted by Frege, are conducted from the logical point of view, without appeal to passages based on intuition; secondly, the proofs of all the major non-arithmetic theorems included by Frege in the first thirteen sections of the Grundgesetze are proposed; finally, some original developments of the series theory and the links between finite series and infinite series are introduced.