Probabilistic databases (PDBs) introduce uncertainty into relational databases by specifying probabilities for several possible instances. Traditionally, they are finite probability spaces over database instances. Such finite PDBs inherently make a closed-world assumption: non-occurring facts are assumed to be impossible, rather than just unlikely. As convincingly argued by Ceylan et al. (KR '16), this results in implausibilities and clashes with intuition. An open-world assumption, where facts not explicitly listed may have a small positive probability can yield more reasonable results. The corresponding open-world model of Ceylan et al., however, assumes that all entities in the PDB come from a fixed finite universe. In this work, we take one further step and propose a model of truly open-world PDBs with an infinite universe. This is natural when we consider entities from typical domains such as integers, real numbers, or strings. While the probability space might become infinitely large, all instances of a PDB remain finite. We provide a sound mathematical framework for infinite PDBs generalizing the existing theory of finite PDBs. Our main results are concerned with countable, tuple-independent PDBs; we present a generic construction showing that such PDBs exist in the infinite and provide a characterization of their existence. This construction can be used to give an open-world semantics to finite PDBs. The construction can also be extended to so-called block-independent-disjoint probabilistic databases. Algorithmic questions are not the focus of this paper, but we show how query evaluation algorithms can be lifted from finite PDBs to perform approximate evaluation (with an arbitrarily small additive approximation error) in countably infinite tuple-independent PDBs.