Chapter 1: Axiomatics, the Social Sciences, and the Gödel Phenomenon: A Toolkit
Partially supported by CNPq, Philosophy Section; the author is a member of the Brazilian Academy of Philosophy. This text was supported in part by CNPq, Philosophy Section, grant no. 4339819902073398. It is part of the research efforts of the Advanced Studies Group, Production Engineering Program, at COPPE–UFRJ and of the Logic Group, HCTE–UFRJ. We thank Profs. A. V. Assumpção, R. Bartholo, C. A. Cosenza, S. Fuks (in memoriam), S. Jurkiewicz, R. Kubrusly, M. Gomes, and F. Zamberlan for support.
The following sections are included:
Introduction
Axiom systems: mathematics
The Gödel phenomenon in physics and in other mathematized sciences
Physics as an archetype for the mathematized sciences
Axiomatics for physics: preliminary steps
Axiomatics for physics: guidelines
Suppes predicates
Axiomatics for physics: the main ideas
Axiomatics for physics: examples
Beyond physics
The incompleteness of analysis
Generalized incompleteness
Higher degrees
θ functions and the arithmetical hierarchy
Statement of the main undecidability and incompleteness results
Questions settled with those techniques
Undecidability and incompleteness in the social sciences
Forcing and our techniques
Evaluation of the results
More on population dynamics
Appendix A. Preliminary remarks
Appendix B. Technicalities
Appendix C. Hard stuff
Appendix D. The counterexample function f
Appendix F. An example
Appendix G. Discussion and more intuitions
Appendix H. An application: Maymin’s theorem
Appendix I. Details
Appendix J. The O’Donnell algorithm
Appendix K. Almost Maymin-efficient markets
Appendix L. A wild idea: inaccessible cardinals?
References
