Is there a name for all numbers greater than zero?
You need to be more specific. If you mean the integers greater than zero, these are called the natural numbers. If you mean real numbers greater than zero, then no, there is no special name for them.
just the use of the adjective positive: "positve reals"
Whether or not the natural numbers contains 0 is a matter of convention.
My general impression is that the convention "0 is a natural number" is becoming standard.
You can also call them the "positive integers".
Separate names with a comma.