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Philosophia Mathematica Advance Access originally published online on January 27, 2007
Philosophia Mathematica 2007 15(1):1-29; doi:10.1093/philmat/nkl028
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Copyright © The Author 2007. Published by Oxford University Press.

Self-reference and the Languages of Arithmetic{dagger}

Richard G. Heck, Jr

Department of Philosophy, Box 1918, Brown University, Providence, R.I. 02912 U.S.A

Correspondence: rgheck{at}brown.edu

I here investigate the sense in which diagonalization allows one to construct sentences that are self-referential. Truly self-referential sentences cannot be constructed in the standard language of arithmetic: There is a simple theory of truth that is intuitively inconsistent but is consistent with Peano arithmetic, as standardly formulated. True self-reference is possible only if we expand the language to include function-symbols for all primitive recursive functions. This language is therefore the natural setting for investigations of self-reference.

Thanks to Albert Visser for extensive, and extremely helpful, comments on an earlier version of this paper. Thanks to John Burgess, Michael Glanzberg, and Vann McGee for discussions of these matters during the long time it took me to sort them out, and to Peter Koellner for helping me refine these ideas during two long sessions in front of his blackboard. Comments by anonymous referees helped clarify the paper at important points.

I presented some of this material as part of a series of seminars I gave in St Andrews in February 2004. Thanks to everyone there, especially AgustFormula n Rayo and Stewart Shapiro, for their comments. That visit was arranged by Crispin Wright and supported by the British Academy and Arché, the AHRC Research Centre for the Philosophy of Logic, Language, Mathematics and Mind. Thanks to both for their support, which is much appreciated.


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