A method of proof

I'm still working on trying to prove the results here with while satisfying as many axioms of conditionals as possible, and ideally with countably additive probabilities. It's not all that easy. Anyway, while thinking about it, I came up with a method of proof that while probably not new, I have never used before.

Suppose we want to close some set A₀ under some finite set of operations with finite arity. One way is to inductively form a sequence of larger and larger sets, A₀⊆A₁⊆A₂⊆..., such that A_n+1 is something like the result of applying the operations to all the tuples of elements of A_n. Then take the union A of the A_n, and that will be our closure, because any finite collection of elements of A will all be in some A_n for a large enough n, and hence the result of applying the operations to that collection will be in A_n+1, and hence in A. So A will be closed under the operations.

But what if our operations have countably infinite arity? An example is probability spaces which we want to be closed under countable unions. In that case, the above method of proof may not be good enough. But there is a cool modification given the Axiom of Choice. By Choice, let β be an ordinal with uncountable cofinality. Use transfinite recursion over α < β to generate a sequence of larger and larger sets A_α, such that A_α is the result of applying the operations to the countable sequences of elements of A_α, and letting A_α, for α a limit ordinal, be the result of applying the operations to the countable sequences of elements of the union of the sets for α'<α. Let A be the union of the A_α. Then a countable sequence of elements of A will all be contained in some one of the A_α because of uncountable cofinality. And then the result of applying the operations to that sequence will be in A_α+1, and hence in A. So A will be closed under the operations.