The difference between Mathematics and Mysticism
There is no "smallest number greater than 0". This is not only because Mathematicians are self-centred egotists, or even because maths profs are Mathematicians (see above) who wish only to flunk their students.
Mathematics believes in doing well-defined things (ONLY). It's practically the science of well-defined things. Having a smallest positive number would break this. For suppose such a number e existed. Then we know e > 0, so e > e/2 > 0, which is a contradiction (e/2 is a smaller positive number!).
So no such thing exists. And you cannot meaningfully "define" something to be this nothing. There is no number 1/infinity, no number 1/aleph0, no number 1/ω, and no number 0.00...001 (where there are "infinitely many" 0's between the decimal point and the 1). None of them make sense, none of them are defined, and all of them are even more nonsensical than the idea of a smallest positive number e as above. And don't get me started on 0.99999...; it really does equal 1 (and no, 1-0.999... is not such a "smallest number greater than 0").
Grow up; deal with it; there are bigger things to worry about.