Constraints on subset sum problem
A set such as $\{1,2,3\}$ is valid input to the problem you defined, so yes, it is "allowed". When setting such constraints, you typically want to know "does the problem still remain NP-complete?". In this sense, notice there is a difference between "can my input take form such as this" and "if my input always takes this form, is the problem still NP-complete?" 

I believe you are after the latter; if you impose some constraints on the input, does the problem remain NP-complete? For example, if your input is the set $\{1,\ldots,n\}$ for some positive integer $n$, the answer is no, the problem can easily be decided in polynomial time (can you see why?)