Feferman’s Skepticism About Set Theory

Chapter
Part of the Outstanding Contributions to Logic book series (OCTR, volume 13)

Abstract

Solomon Feferman has expressed skepticism or reserve about set theory, especially higher set theory, in many writings, and in his mathematical work he has largely stayed away from set theory. The paper undertakes to describe and diagnose Feferman’s attitude toward set theory, especially higher set theory. Section 1 discusses his opposition to Platonism in relation to some understandings of what Platonism is. Section 2 discusses his interest in predicativity, his analysis of predicative provability, and his sympathy for “predicativism,” although he denies being an adherent and has used impredicative methods in his own metamathematical work. It also notes his reconstruction of Hermann Weyl’s attempt to construct the elements of analysis on a predicative basis. Section 3 concerns his attitude toward proof theory. Section 4 turns more to philosophy. His anti-platonism is compared with the well known platonism of Gödel. Unlike Gödel, Feferman views concepts as human creations. He notes that basic mathematical concepts can differ in clarity and argues that less clear concepts, in particular those of set theory, can give rise to questions that do not have definite answers. It is questioned whether Feferman’s conceptual structuralism gives mathematics the degree of objectivity that its application in science requires. In Sect. 5 remarks are made about Hilary Putnam’s criticism of Feferman’s claim that a predicative system conservative over PA is adequate for the mathematics applied in science. The difference is seen to turn on Putnam’s scientific realism.

Keywords

Logic Set theory Predicativity Proof theory Skepticism 

Subject Classification

Logic and foundations 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of PhilosophyHarvard UniversityCambridgeUSA

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