  Exploring further the properties of ITRM-recognizable reals, we provide a
detailed analysis of recognizable reals and their distribution in G\"odels
constructible universe L. In particular, we show that, for unresetting infinite
time register machines, the recognizable reals coincide with the computable
reals and that, for ITRMs, unrecognizables are generated at every index bigger
than the first limit of admissibles. We show that a real r is recognizable iff
it is $\Sigma_{1}$-definable over $L_{\omega_{\omega}^{CK,r}}$, that $r\in
L_{\omega_{\omega}^{CK,r}}$ for every recognizable real $r$ and that either all
or no real generated over an index stage $L_{\gamma}$ are recognizable.
