What is Absolute Undecidability?†
Article first published online: 21 JUN 2012
© 2012 Wiley Periodicals, Inc.
Volume 47, Issue 3, pages 467–481, September 2013
How to Cite
Clarke-Doane, J. (2013), What is Absolute Undecidability?. Noûs, 47: 467–481. doi: 10.1111/j.1468-0068.2012.00861.x
- Issue published online: 10 JUL 2013
- Article first published online: 21 JUN 2012
Vol. 48, Issue 4, 809, Article first published online: 27 OCT 2014
It is often alleged that, unlike typical axioms of mathematics, the Continuum Hypothesis (CH) is indeterminate. This position is normally defended on the ground that the CH is undecidable in a way that typical axioms are not. Call this kind of undecidability “absolute undecidability”. In this paper, I seek to understand what absolute undecidability could be such that one might hope to establish that (a) CH is absolutely undecidable, (b) typical axioms are not absolutely undecidable, and (c) if a mathematical hypothesis is absolutely undecidable, then it is indeterminate. I shall argue that on no understanding of absolute undecidability could one hope to establish all of (a)–(c). However, I will identify one understanding of absolute undecidability on which one might hope to establish both (a) and (c) to the exclusion of (b). This suggests that a new style of mathematical antirealism deserves attention—one that does not depend on familiar epistemological or ontological concerns. The key idea behind this view is that typical mathematical hypotheses are indeterminate because they are relevantly similar to CH.