symmetry and the brown-freiling refutation of the continuum hypothesis对称和brown-freiling连续统假设的驳斥.pdf

Symmetry 2011, 3, 636-652; doi:10.3390/sym3030636 OPEN ACCESS symmetry ISSN 2073-8994 /journal/symmetry Article Symmetry and the Brown-Freiling Refutation of the Continuum Hypothesis Paul Bartha Department of Philosophy, University of British Columbia, Vancouver, BC V6T 1Z1, Canada; E-Mail: pbartha@exchange.ubc.ca; Tel.: 604-822-2621; Fax: 604-822-8782 Received: 25 July 2011; in revised form: 27 August 2011 / Accepted: 1 September 2011 / Published: 6 September 2011 Abstract: Freiling [1] and Brown [2] have put forward a probabilistic reductio argument intended to refute the Continuum Hypothesis. The argument relies heavily upon intuitions about symmetry in a particular scenario. This paper argues that the argument fails, but is still of interest for two reasons. First, the failure is unusual in that the symmetry intuitions are demonstrably coherent, even though other constraints make it impossible to find a probability model for the scenario. Second, the best probability models have properties analogous to non-conglomerability, motivating a proposed extension of that concept (and corresponding limits on Bayesian conditionalization). Keywords: symmetry; probability; Continuum Hypothesis; conglomerability; finitely additive measures; paradoxical sets 1. Introduction In the context of his work on thought experiments, Brown [2–4] discusses a remarkable argument meant to refute the Continuum Hypothesis (CH). The argument takes the form