Paradoxes, Self-Reference and Truth in the 20th Century

The general aim of the chapter is to describe the emergence of non-trivial logical themes from the discussion on paradoxes, with a special emphasis on the semantical paradoxes and in connection with the foundations of property theory and set theory. The present historical analysis mainly deals with logical works, which focus on three interacting key-notions: truth, self-reference, type free comprehension. The basic questions we try to answer are: 1. how does the question of paradoxes relate to the main stream of mathematical logic (at different stages of development)? 2. Which relevant logical tools emerge and/or are refined, in strict connection with the discussion of paradoxes about truth and predicate application? As a final outcome, we would like to reach a reasonably complete catalogue of ideas, methods and logical tools, involved in the investigation of paradoxes which took place in the past century. It should also come into view that the role of semantical paradoxes for a modern mathematical development of definability theory (in its various form: recursion theory, descriptive set theory) is certainly not negligible and it arises in close correlation with set theory, model theory, recursion theory.