Nested random intercept I am calculating a three level HLM. The first level are the measurement points, the second are subjects and the third therapists. I am investigating the effect of an intervention, which is only delivered to half of the participants - which again means that only half of them are nested within therapists. I assume intercepts to vary across subjects, which again are nested within therapists. I know how to do this i R, but I was wondering about the exact mathematical equation for random nested intercepts.
This would be my level 1 equation: 
$$Y_{ijk}= β_{0jk}+ β_{1jk}Time_{ijk}+R_{ijk}$$
with i indicating the measurement point, j the subject and k the therapist
Than level 2 would be: 
$$β_{0jk}= γ_{00k}+ γ_{01}Treatment_{jk}+ U_{0jk}$$
$$β_{1jk}= γ_{10k}+ γ_{11}Treatment_{jk}$$
For level 3, I thought of this one:
$$γ_{00k}= δ_{000}+V_{0k}$$
$$γ_{10k}= δ_{110}$$
However, I was wondering wether thiese equations really represent a nested design or wether this is simply estimating a deviation of the intercept for subjects and therapists, but not for subjects nested within therapists. 
 A: Your equations look OK, except that in the first equation for level 3, I think $V_{0k}$ should be $V_{00k}$. 
Moving on to the substance of the question:

I was wondering wether thiese equations really represent a nested design or wether this is simply estimating a deviation of the intercept for subjects and therapists, but not for subjects nested within therapists.

Yes, you are using standard notation found in the multilevel modelling literature for nested factors.
In terms of notation, the crux of the matter comes down to the index $j$ for subjects. If subjects are actually nested within therapists, it means that each subject "belongs" to one and only one therapist, so that in turn means that if there are $n$ subjects in total, then $j \in [1, \dots, n]$ and this is how the multilevel modelling literature is usually framed. 
If subjects are not nested in therapists, then it means they are crossed, so each subject "belongs" to more than one therapist. If they are fully crossed then each subject belongs to all therapists. In other words all subjects are treated by all therapists. In that case we would have $j \in [1, \ldots, n_p]$ for each of $p$ therapists. So in the simple case of 2 therapists and 2 subjects, therapist 1 would treat subject 1 and subject 2, and therapist 2 would also treat subject 1 and subject 2.
