Combinatorial principles like inclusion-exclusion can be obtained by decategorifying natural structures on the category of finite sets, whose Grothendieck ring is the ring of integers, Z. In this talk, we explain what happens when similar structures are decategorified from the category of almost-finite cyclic sets, whose Grothendieck ring is the big Witt ring, W(Z). A natural example of an almost-finite cyclic set is the set of \overline{Fq}-points of an algebraic variety over Fq, and the corresponding element in W(Z) is the zeta function of the variety; using this, we explain how combinatorics in the ring of big Witt vectors can be applied to prove theorems in arithmetic and motivic statistics.