A note on Gödel’s philosophy of mathematics in the light of his Nachlass

The problem of giving a coherent unifying picture for all aspects of Gödel's philosophy of mathematics, still appears as an appealing task. The present research, is intended as a contribution to this direction of work. We start by shortly summarizing Gödel's philosophy of mathematics, focusing the attention on the crucial role of the act of understanding abstract concepts. We then offer a possible explanation of Gödel's position, trying to make clear its relation with Gödel's reflections on "axioms of infinity'' as possibile ways to significantly extend our set­theoretical knowledge. In other words, we try to give an answer to the following question: Why did Gödel insist on similar principles, while stressing their unusefulness to solve the problem (decidability of Cantor's Continuum Hypothesis) which primarily indicated the need for the above mentioned extension? Two remarks turn out to be of special importance: (i) these axioms stem from the very concept (the iterative conception of set) which is at the base of a ``truly mathematical'' theory of the informal notion of ``aggregate''; (ii) they make decidable otherwise undecidable Diophantine propositions. We give an analysis of how Gödel argumented in favor of these reasons, thus emphasizing, in particular, a relation with Gödel's reflections on the significance of the incompleteness theorems. We make an essential use of only recently published material coming from Gödel's Nach­lass, which helps us to recast some of his known papers in an hopefully new perspective.