not to muddy the waters, or anything (heh), but:

"Signed zero is zero with an associated sign. In ordinary arithmetic, −0 = +0 = 0. However, in computing, some number representations allow for the existence of two zeros, often denoted by −0 (negative zero) and +0 (positive zero)."
-wiki

so, you may say that 0 is neither positive or negative (in your set theory discussions); or you may say zero is *both positive and negative (in your floating point discussions).

also, consider this statement: zero is an even integer, although it is neither positive nor negative.