What is the set of central charges of two dimensional rational conformal field theories?

Question

In this question,  RCFT means a unitary full rational two dimensional conformal field theory. As every unitary two dimensional conformal field theory, a RCFT has a central charge $c$ which is a nonnegative real number. The rationality hypothesis implies that in fact $c$ is a nonnegative rational number: $c \in \mathbb{Q}_{\geq 0}$.   

Let $\mathcal{C}$ be the subset of $\mathbb{Q}_{\geq 0}$ made of rational numbers which are central charge of some RCFT. As it is possible to tensorize RCFTs, $\mathcal{C}$ is an additive subset of $\mathbb{Q}_{\geq 0}$.

For example, the intersection of $\mathcal{C}$ with the interval $[0,1]$ is given by $0$, $1/2$, $7/10$, $4/5$,...., $1$ i.e. by the central charges of the unitary minimal models union $c=1$. In particular, $c=1$ is an accumulation point of $\mathcal{C}$.

My questions are: 

Quantitative: Is the set $\mathcal{C}$ explicitely known ?

Qualitative: Is the set $\mathcal{C}$ closed in $\mathbb{R}$ ? Is it well-ordered ? If yes, what is its ordinal ? (for example, are there accumulation points of accumulation points...)

These questions have two motivations:

1) the claim that the RCFT's are classified: see for example http://ncatlab.org/nlab/show/FRS-theorem+on+rational+2d+CFT  

I did not go through this work but I would like to know if this classification is "abstract" or "concrete". In particular, I would like to know if it gives an answer to the previous questions.

2)Similar questions have been asked and solved for a different set of real numbers: the set of volumes of hyperbolic 3-manifolds. It seems to me that there is a (very vague at this moment) similarity between these two sets of real numbers.

Comments

I think above 1, the spectrum is continuous, but I need to check the yellow book.

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@Ryan Thorngren : as I am restricting myself to rational CFTs, the set of central charges is certainly not continuous. But even considering all the CFTs, I don't think it is true. For example, the existence of some CFTs with irrational central charges was a not so easy question as far as I understand. Are you rather refering to the fact that there exists unitary representations of the Virasoro algebra for any value of the central charge above 1?