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Philosophia Mathematica Advance Access originally published online on January 9, 2006
Philosophia Mathematica 2006 14(2):134-152; doi:10.1093/philmat/nkj003
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Philosophia Mathematica (III), Vol. 14 No. 2 © The Author [2006]. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: journals.permissions@oxfordjournals.org

Are There Absolutely Unsolvable Problems? Gödel's Dichotomy{dagger}

Solomon Feferman*

* Departments of Mathematics and Philosophy, Stanford University Stanford, California 94305, U. S. A. sf{at}csli.stanford.edu

This is a critical analysis of the first part of Gödel's 1951 Gibbs lecture on certain philosophical consequences of the incompleteness theorems. Gödel's discussion is framed in terms of a distinction between objective mathematics and subjective mathematics, according to which the former consists of the truths of mathematics in an absolute sense, and the latter consists of all humanly demonstrable truths. The question is whether these coincide; if they do, no formal axiomatic system (or Turing machine) can comprehend the mathematizing potentialities of human thought, and, if not, there are absolutely unsolvable mathematical problems of diophantine form.

Either ... the human mind ... infinitely surpasses the powers of any finite machine, or else there exist absolutely unsolvable diophantine problems.


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