Defining Completeness in the Theory of Boolean Algebra

Question

If one were formalising the theory of Boolean Algebra in first-order logic, would it be possible to define completeness with binary meets and joins, i.e., somehow allowing for infinite meets and joins?

Answer 1

No, this can't be done.

It certainly can't be done directly: the syntax of first-order logic doesn't treat infinite-arity operations, so a complete Boolean algebra thought of as a set $B$ together with a unary function (negation) and two set-ary functions (arbitrary meet and arbitrary join) isn't even a first-order structure.

Less trivially, it can't be done indirectly, either: there is no set $\Gamma$ of first-order sentences in the language of Boolean algebras which is true in exactly the complete Boolean algebras. The quicket way to see this is via the downward Lowenheim-Skolem theorem: the powerset of $\mathbb{N}$, viewed as a Boolean algebra in the natural way, is complete, but no countable elementary subalgebra of it is complete.

---

Re: the first point above, it's worth noting that we could of course consider an expansion of first-order logic to handle infinite-arity operations. This (or rather, these - there are multiple ways to implement the idea) is surprisingly poorly behaved, and even the tamest approaches lack both the compactness and Lowenheim-Skolem properties.