A Note on Infinities in Eternal Inflation
  In some well-known scenarios of open-universe eternal inflation, developed by
Vilenkin and co-workers, a large number of universes nucleate and thermalize
within the eternally inflating mega-universe. According to the proposal, each
universe nucleates at a point, and therefore the boundary of the nucleated
universe is a space-like surface nearly coincident with the future light cone
emanating from the point of nucleation, all points of which have the same
proper-time. This leads the authors to conclude that at the proper-time t =
t_{nuc} at which any such nucleation occurs, an infinite open universe comes
into existence. We point out that this is due entirely to the supposition of
the nucleation occurring at a single point, which in light of quantum cosmology
seems difficult to support. Even an infinitesimal space-like length at the
moment of nucleation gives a rather different result -- the boundary of the
nucleating universe evolves in proper-time and becomes infinite only in an
infinite time. The alleged infinity is never attained at any finite time.
