Infinity, Causation, and Paradox

Abstract

Infinity is paradoxical in many ways. A particular large family of paradoxes is examined that on its face iswidely varied. Some involve deterministic supertasks, such as Thomson’s Lamp where a switch is toggled an infinite number of times over a finite period of time, or the Grim Reaper, where it seems that infinitely many reapers can produce a result without doing anything. Others involve infinite lotteries. Yet others involve paradoxical results in decision theory, such as the surprising observation that if you perform a sequence of fair coin-flips that goes infinitely far back into the past but only finitely into the future, you can leverage information about past coin-flips to predict future ones with only finitely many mistakes. It turns out that these, and a number of other paradoxes have a common structure: their most natural embodiment involves an infinite number of items causally impinging on a single output. These paradoxes can all be solved with a single move: embrace causal finitism, the view that it is impossible for a single output to have an infinite causal history. The book exposits such paradoxes, defends causal finitism at length, and ends up considering connections with the philosophy of physics, where causal finitism favors, but does not require, discretist theories of space and time, and the philosophy of religion, where we get a cosmological argument reminiscent of the Kalām argument for the existence of God.