Am I right to assume that under the frequentist interpretation of probability,* the set of allowed probabilities isn't $$\left[0,1\right],$$ but rather is $$\lim_{m\to\infty}\left.\left\{\frac{n}{m}\,\right|\; n=0,\ldots,m\right\}$$ or perhaps $$\bigcup_{m=1}^\infty\left.\left\{\frac{n}{m}\,\right|\; n=0,\ldots,m\right\},$$ or $$\left[0,1\right]\cap\Bbb Q,$$ all implying, I assume, e.g., i) that the set of allowed probabilities is countable, and ii) that $\frac1\pi$ isn't an allowed probability?
NB: Besides a straight answer, a little background or some references would be highly appreciated.
*Perhaps, or perhaps not, including that "[a] controversial claim of the frequentist approach is that in the "long run," [sic] as the number of trials approaches infinity, the relative frequency will converge exactly to the true probability [...]".