I have been studying quantum mechanics, specifically angular momentum, but I have a question that concerns raising and lowering operators as a whole. For total angular momentum, you can define: $$J_\pm=J_x\pm iJ_y $$ Anyone who is familiar with angular momentum will recognize these as the raising and lowering operators, but I will continue on with the problem to better explain my question.
An analysis of this problem shows that: $$ [J_z, J_\pm]=\pm \hbar J_\pm$$ $$ [J^2, J_\pm]=0 $$ From here it is easy to see that if $J_z|\alpha\beta\rangle= \beta|\alpha\beta\rangle, $ and $J^2|\alpha\beta\rangle= \alpha|\alpha\beta\rangle$, $$ J_z(J_+|\alpha\beta\rangle)=(J_+J_z+\hbar J_+)|\alpha\beta\rangle= (J_+\beta+\hbar J_+)|\alpha\beta\rangle=(\beta +\hbar)J_+|\alpha\beta\rangle $$ And thus we can say $J_+|\alpha\beta\rangle=C|\alpha,\beta + \hbar\rangle $.
However, while this approach is very clean cut, in my mind it doesn't exactly show that the eigenvalues of $J_z$ exist only in increments of $\hbar$. For instance, if I were able to find some arbitrary set of operators $W_\pm$, such that $[J_z, W_\pm]=\pm (\hbar /4)W_\pm$, then I could easily show by the logic above that the eigenvalues of $J_z$ exist in increments of $\hbar /4$. So then, what guarantees that I cannot find such operators? More specifically, what part of the "raising and lowering operator" method guarantees that there are not more possible eigenvalues of $J_z$ (or any operator), than those found using raising and lowering operators?