>>8 It depends on what form of logic you use. Some more advanced types require the set to be non empty, so that logical consequences such as
Ax(Px)->Ex(Px)
"(For all x, x has some property P) implies (There exists an x, such that x has some property P)"
are possible to use, and so in light of those styles of logical proofs you must make the initial assumption that what you're talking about exists.
>>1 may not have made the assumption that maths exists, which is perfectly fine and your argument doesn't quite apply.