  Infinite Time Register Machines ($ITRM$'s) are a well-established machine
model for infinitary computations. Their computational strength relative to
oracles is understood, see e.g. Koepke (2009), Koepke and Welch (2011) and
Koepke and Miller (2008). We consider the notion of recognizability, which was
first formulated for Infinite Time Turing Machines in Hamkins and Lewis (200)
and applied to $ITRM$'s in Carl et al. (2010). A real $x$ is
$ITRM$-recognizable iff there is an $ITRM$-program $P$ such that $P^{y}$ stops
with output 1 iff $y=x$, and otherwise stops with output 0. In Carl et al.
(2010), it is shown that the recognizable reals are not contained in the
computable reals. Here, we investigate in detail how the $ITRM$-recognizable
reals are distributed along the canonical well-ordering $<_{L}$ of G\"odel's
constructible hierarchy $L$. In particular, we prove that the recognizable
reals have gaps in $<_{L}$, that there is no universal $ITRM$ in terms of
recognizability and consider a relativized notion of recognizability.
