# 22.15 Universal Positivity Set

Can one construct an infinite recursively enumerable set $$S\subset \mathbb{N}$$, for which one can decide: given any linear recurrence sequence $$(u_n)$$, whether $$\exists n\in S$$, s.t. $$u_n<0$$ ? < p>

In other words: is there a set of indices for which the positivity problem is decidable?

The analogous question where $$u_n<0$$ is replaced by $$u_n="0$$" has a positive answer for simple sequences; luca, ouaknine and worrell have constructed such set explicitly (called the universal skolem set). < p>