Almost Every Domain is Universal

We endow the collection of ω-bifinite domains with the structure of a probability space, and we will show that in this space the collection of all universal domains has measure 1. For this, we present a probabilistic way to extend a finite partial order by one element. Applying this procedure iteratively, we obtain an infinite partial order. We show that, with probability 1, the cpo-completion of this infinite partial order is the universal homogeneous ω-bifinite domain. By alternating the probabilistic one-point extension with completion procedures we obtain almost surely the universal and homogeneous ω-algebraic lattice, ω-Scott domain, and ω-bifinite L-domain, respectively. We also show that in the projective topology, the set of universal and homogeneous ω-bifinite domains is residual (i.e., comeagre), and we present an explicit number-theoretic construction of such a domain.