Chapter

Which Undecidable Mathematical Sentences Have Determinate Truth Values?

Hartry Field

in Truth and the Absence of Fact

Published in print March 2001 | ISBN: 9780199242894
Published online November 2003 | e-ISBN: 9780191597381 | DOI: http://dx.doi.org/10.1093/0199242895.003.0012
 Which Undecidable Mathematical Sentences Have Determinate Truth Values?

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Argues that typical undecidable sentences of set theory (e.g. about the size of the continuum) can have no determinate truth‐value, since nothing in our practice can determine which ‘universe of sets’ we are talking about. But it gives an account of how typical undecidable number‐theoretic sentences can have determinate truth‐value. The account has it though, that, whether these undecidable number‐theoretic sentences have determinate truth‐value, turns on assumptions about the physical world that could fail, and it explores the consequence of their failure. It concludes with a critique of an argument that at least the Gödel sentence of our fullest mathematical theory must be determinately true: the critique is that the argument depends on the same sort of reasoning that leads to the semantic paradoxes.

Keywords: continuum hypothesis; finiteness; Gödel; incompleteness theorem; indeterminacy; objectivity; second‐order logic; semantic paradoxes; truth; undecidable sentences

Chapter.  15550 words. 

Subjects: Philosophy

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