Optimal results on ITRM-recognizability

Abstract

Exploring further the properties of ITRM-recognizable reals, we provide a detailed analysis of recognizable reals and their distribution in Gödels constructible universe L.In particular, we show that new unrecognizables are generated at every index $\gamma\geq\omega_{\omega}^{CK}$. We give a machine-independent characterization of recognizability by proving that a real $r$ is recognizable iff 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$ITRM$s is recognizable and that the set of $ITRM$-computable reals is not $ITRM$-decidable.