Is the language $L_{universal} = \{ \left \langle M \right \rangle | M \textrm{is a universal turing machine} \}$ decidable?
I'm guessing it is decidable according to the definition of a UTM, that a UTM must be able to calculate every recursive function. Since the set of recursive languages and the set of all input words are both enumerable, we are theoretically able to determine if the given $\left \langle M \right \rangle$ is a UTM. Is my logic somewhat correct?