Are There Absolutely Unsolvable Problems? Godel's Dichotomy

Philosophia Mathematica 14 (2):134-152 (2006)
This is a critical analysis of the first part of Go¨del’s 1951 Gibbs lecture on certain philosophical consequences of the incompleteness theorems. Go¨del’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
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DOI 10.1093/philmat/nkj003
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