This initial chapter is divided into two sections. The first is devoted to a brief exposition of the intuitive essence and the philosophical motivation of Gödel’s main metamathematical results, namely his completeness theorem for elementary logic (1930) and his incompleteness theorems for arithmetic (1931). Thereafter some discussion of the different ways to confront the relationship between those results and Gödel’s philosophical realism in logic and mathematics is offered. Thus, mathematical realism will be successively regarded as (i) a philosophical consequence of those results; (ii) a heuristic principle which leads to them; (iii) a philosophical hypothesis which is “verified” by them. In the second section Gödel’s philosophy of mathematics, such as it can be derived from his published writings, is briefly expounded upon. Then the final version of his essay on Carnap is summed up, in order to see how his unpublished philosophical ideas might throw some light on Gödel’s published doctrines. Finally, other relevant ideas and authors are briefly surveyed.
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