An Infinite Lottery Paradox

Abstract

In a fair, infinite lottery, it is possible to conclude that drawing a number divisible by four is strictly less likely than drawing an even number; and, with apparently equal cogency, that drawing a number divisible by four is equally as likely as drawing an even number.

Strict dominance and symmetry

“…Thus, a member of G is a permutation ρ n , for an integer n, such that ρ n m = n + m. In this case, the Axiom of Choice is not needed, as we can exhibit the wager V explicitly as V (n) = φ(n) for any bounded strictly increasing function φ. But while this example is mathematically trivial, the possibility of infinite fair lotteries is quite philosophically controversial (see Pruss [12,Chapter 4] and Norton and Parker [7]).…”

“…6 And it is plausible that non-measurable functions are not implementable as actual games. 7 Formally, we then have this fact contrasting with Proposition 1:…”

“…(6) Ideal agents' preferences always satisfy Strict Dominance. (7) Ideal agents' preferences are sometimes invariant under symmetries that fail condition (iv) of Proposition 1, such as rotational symmetry for uniform spinners or translational symmetry for infinite countable fair lotteries. I will take the Axiom of Choice for granted-it is widely accepted by mathematicians, and a serious discussion would move us from decision theory to the philosophy of mathematics.…”

“…But that's enough for the integral to be strictly positive. 7 Solovay [14] has famously shown that assuming a certain large cardinal assumption, the existence of a non-measurable set requires some version of the Axiom of Choice. If the assumption holds, then any paradoxical wager like the one in the spinner case requires the Axiom of Choice.…”

Are there Mathematical Thought Experiments?

“…356-366;Sherry 2006;Shaffer 2015;Hertogh 2021. More generally, on mathematical TEs, see above all Witt-Hansen 1976;Mueller 1969;Brown 1999Brown , 2004Brown , 2007Brown , 2011Brown , 2017Brown , and 2021Glas 2001aand 2001b, Van Bendegem 2003Buzzoni 2004Buzzoni , 2008Buzzoni , 2011Buzzoni , 2021aBuzzoni and 2021bSherry 2006;Starikova 2007;Cohnitz 2008;Starikowa and Giaquinto 2018, Brown 2022, Norton & Parker 2022, Lenhard 2022, Fehige and Vestrucci 2022 rejected because of a simple but decisive objection: it is a fact that many authors, among them some mathematicians, have spoken about TEs in mathematics, illustrating this meaning of TE with concrete examples. This, however, in spite of undermining my earlier conclusion, does not force one to disavow all the reasons that led to it.…”