Exploring further the properties of ITRM-recognizable reals started in [1], we provide a detailed analysis of recognizable reals and their distribution in Gödels constructible universe L. In particular, we show that new unrecognizable reals are generated at every index
$\gamma \, \ge \,\omega _\omega ^{CK}$
. We give a machine-independent characterization of recognizability by proving that a real r is recognizable if and only if it is
${\Sigma _1}$
-definable over
${L_{\omega _\omega ^{CK,\,r}}}$
and that
$r\, \in \,{L_{\omega _\omega ^{CK,\,r}}}$
for every recognizable real r and show that either every or no r with
$r\, \in \,{L_{\omega _\omega ^{CK,\,r}}}$
generated over an index stage
${L_\gamma }$
is recognizable. Finally, the techniques developed along the way allow us to prove that the halting number for ITRMs is recognizable and that the set of ITRM-computable reals is not ITRM-decidable.