Symmetry and the Brown-Freiling refutation of the continuum hypothesis (Q527582)

Summary: \textit{C. Freiling} [J. Symb. Log. 51, 190--200 (1986; Zbl 0619.03035)] and \textit{J. R. Brown} [Philosophy of mathematics. A contemporary introduction to the world of proofs and pictures. 2nd ed. London: Routledge (2008; Zbl 1171.00006)] have put forward a probabilistic \textit{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).