2. "Constructive truth and circularity". I propose a constructive interpretation of self-applicative truth and use it to resolve the standard semantic paradoxes. This paper can be read independently of the others. In LaTex, DVI, PS, PDF.
3. "Mathematical conceptualism". An explanation and defense of conceptualism for a general mathematical and philosophical audience. Start here. In LaTex, DVI, PS, PDF.
4. "Is set theory indispensable?" This is my most thorough explanation of why I feel set theory is not an appropriate foundation for mathematics. Written for a general mathematical and philosophical audience. In LaTex, DVI, PS, PDF.
5. "Analysis in J_2". This is an expository paper in which I explain how core mathematics, particularly abstract analysis, can be developed within a concrete countable set J_2 (the second set in Jensen's constructible hierarchy). The implication, well-known to proof theorists but probably not to most mainstream mathematicians, is that ordinary mathematical practice does not require an enigmatic metaphysical universe of sets. I go further and argue that J_2 is a superior setting for normal mathematics because it is free of irrelevant set-theoretic pathologies and permits stronger formulations of existence results. In LaTex, DVI, PS, PDF.
6. "The concept of a set". I propose that there are actually three distinct notions of a "collection" (surveyable, definite, and heuristic) that appear in mathematics, and that they are governed by three different kinds of logic (classical, intuitionistic, and minimal). In LaTex, DVI, PS, PDF.
7. "Axiomatizing mathematical conceptualism in third order arithmetic". In "Analysis in J_2" I show how ordinary mathematics can be done within the setting of a countable structure J_2 which plays the role of a miniature set-theoretic universe. Here I develop an axiomatic approach to formalizing mathematics that directly expresses the basic principles of conceptualism. In LaTex, DVI, PS, PDF.
8. "Predicativity beyond Gamma_0". I reevaluate the claim that predicative reasoning (given the natural numbers) is limited by the Feferman-Schutte ordinal Gamma_0. First I comprehensively criticize the arguments that have been offered in support of this position. Then I analyze predicativism from first principles and develop a general method for accessing ordinals which is predicatively valid according to this analysis. I find that the Veblen ordinal \phi_{\Omega^\omega}(0), and larger ordinals, are predicatively provable. In LaTex, DVI, PS, PDF.
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Solomon Feferman has given a detailed critique of the paper "Predicativity beyond Gamma_0", and I have written a comprehensive response to his critique.
Solomon Feferman's response (posted here with
permission)
My response to Solomon Feferman's letter
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Last modified May 11, 2009
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