1. "Truth and the liar paradox". We analyze the informal notion of truth and conclude that it can be formalized in essentially two distinct ways: constructively, in terms of provability, or classically, as a hierarchy of concepts which satisfy Tarski's biconditional in limited settings. This leads to a complete resolution of the liar paradox. In LaTex, PS, PDF.

2. "The semantic conception of proof". We analyze the informal semantic conception of proof and axiomatize the proof relation and the provability operator. A self referential propositional calculus which admits provable liar type sentences is introduced and proven consistent. We also investigate the problem of interpreting arbitrary formal systems in systems which include a provability operator. In LaTex, PS, PDF.

3. "Reasoning about constructive concepts". We find that second order quantification is problematic when a quantified concept variable is supposed to function predicatively. This issue is analyzed and it is shown that a constructive interpretation of the falling under relation suffices to resolve the difficulty. We are then able to present a formal system for reasoning about concepts. We prove that this system is consistent and we investigate the extent to which it is able to interpret set theoretic and number theoretic systems of a more standard type. In LaTex, PS, PDF.

4. "Kinds of concepts". The central focus is on clarifying the distinction between sets and proper classes. To this end we identify several categories of concepts (surveyable, definite, indefinite), and we attribute the classical set theoretic paradoxes to a failure to appreciate the distinction between surveyability and definiteness. In LaTex, PS, PDF.

5. "What is predicativism?" A survey. In LaTex, PS, PDF.

6. "Intuitionism and the liar paradox". I discuss the liar type sentence "This sentence is not provable" in the context of intuitionism. A less complete and less formal treatment of the material in "Truth and the liar paradox". In LaTex, PS, PDF.

7. "Constructive truth and circularity". An early exposition of the ideas that were more fully expressed later in papers 1, 2, and 6. In LaTex, PS, PDF.

8. "Mathematical conceptualism". An explanation and defense of conceptualism for a general mathematical and philosophical audience. In LaTex, PS, PDF.

9. "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, PS, PDF.

10. "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, PS, PDF.

11. "The concept of a set". This is superseded by "Kinds of concepts". In LaTex, PS, PDF.

12. "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, PS, PDF.

13. "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, 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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