Infinite Time and the Boltzmann Brain Hypothesis

Many argue that the standard understanding of the second law of thermodynamics combined with the supposition, backed by recent scientific evidence, that the future is infinite entails that one is, most likely, a momentary Boltzmann brain that will quickly disintegrate into the cosmos. The argument is as follows: (1) Given infinite time, the universe will eventually reach thermodynamic equilibrium; (2) once there, every possible fluctuation away from equilibrium, no matter how improbable, will recur, ad infinitum; (3) those fluctuations that create stable, long-lived creatures, such as we take ourselves to be, will be extremely rare compared to those that create short-lived brains that mistakenly think they are ordinary human beings; hence, by statistical reasoning, (4) one is, with overwhelming probability, just a fleeting instantiation of experience. I argue that this reasoning is invalid since it rests on an error regarding the relationship between infinite sets and their subsets. Once this error is eliminated, the power of the argument fades, and the evidence that we are ordinary human beings becomes decisive. Surprisingly, I find that the best argument for the Boltzmann brain hypothesis requires the assumption that the future is very long but finite.