I don't think OP was saying that "for all" is any more meaningful from a computational perspective than "there exits". The point is that both are higher order functions that return an answer for any function on all natural numbers (or inputs in a more general setting).
A set doesn't need to be infinite for us to not be able to say something about all it's members. Each busy beaver number is very much finite but you'd run out of universe if you tried calculating BB value for a number as small as 12.
Whether something is practical to compute is a very different matter from what OP stated was their concern, namely logical knots, potential inconsistencies/contradictions of the "this sentence is false" type, and unbounded complexity.
It really isn't. Math proofs need to be not only finite but small enough to fit in the universe to be discovered. This strictly finite nonsense is a left over from a time when we could only do symbol manipulation with a human brain.
