We take an ordered ring to be a structure of type $(+ - \times < 0\,\, 1)$ satisfying the usual axioms. If $A$ is an ordered ring then we say that an element $a$ of $A$ is infinitesimal if for all integers $n$ it holds that $-1<na<1$.

Let $T$ be the set of sentences that hold in every ordered ring that has no infinitesimal elements other than 0.

Question: Is $T$ effectively enumerable?