We consider the number of independent sets in hypergraphs, which allows us to define the independence density of countable hypergraphs. Hypergraph independence densities include a broad family of densities over graphs and relational structures, such as F-free densities of graphs for a given graph F. In the case of k-uniform hypergraphs, we prove that the independence density is always rational. In the case of finite but unbounded hyperedges, we show that the independence density can be any real

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... number in [0,1]. Finally, we extend the notion of independence density via independence polynomials.