The probability of an infinite sequence of heads is zero

Williamson (2007) has given a celebrated isomorphism or translation-invariance argument that an infinite sequence of heads should have zero probability. However, infinitesimalists who hold that such a sequence has non-zero infinitesimal probability are very likely to deny the translation-invariance (e.g., Weintraub, 2008). Thus, we give two new arguments that an infinite sequence of heads has zero probability, without relying on the full translation-invariance thesis. We also show that two of our arguments extend to the setting of primitive conditional probabilities such as Popper functions and to total comparative probabilities.