I know exactly what the axioms for a semiring of sets are, but I was looking for a way to write them in formal logic. I'm especially pointing to the third axiom asserting the existence of a finite sequence of disjoint sets from the semiring itself. Can you help?
Thanks as usual! :)
The answer is that it is impossible to formalize the desired axiomatization in FOL, just as it is impossible to axiomatize the reals in FOL. You can do this in SOL (second-order logic) with full semantics, but of course the consequence is that what are models of the resulting axiomatization would be sensitive to the meta-system.
In particular, we can prove the claimed impossibility by compactness of FOL. Take any FOL theory $S$ that axiomatizes only semi-ring families. Let $L$ be the language of $S$ plus new constant-symbols $A,B$. For each natural $k$, let $φ_k$ be an FOL sentence over $L$ that states ( $A ∖ B$ is not a union of $k$ members of the family ). Note that $φ_k$ needs $k$ quantifiers. But for each natural $k$ there is a model $M_k$ of $S ∪ \{ φ_i : 0≤i≤k \}$, namely the family $\{ \varnothing \} ∪ \{ \{x\} : x∈[0..k] \} ∪ \{ [0..k] \}$, since $[0..k] ∖ \varnothing$ is not the union of $k$ members of the family. Thus by compactness $S ∪ \{ φ_i : i∈\mathbb{N} \}$ is consistent and has a model, contradicting the definition of $S$.