Optimal results on ITRM-recognizability

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CARL, Merlin, 2013. Optimal results on ITRM-recognizability

@unpublished{Carl2013Optim-25580, title={Optimal results on ITRM-recognizability}, year={2013}, author={Carl, Merlin} }

Optimal results on ITRM-recognizability terms-of-use 2014-01-11T13:36:47Z Carl, Merlin 2014-01-11T13:36:47Z Exploring further the properties of ITRM-recognizable reals, we provide a detailed analysis of recognizable reals and their distribution in Gödels constructible universe L.<br />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 $L_{\omega_{\omega}^{CK,r}}$<br />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<br />$ITRM$s is recognizable and that the set of $ITRM$-computable reals is not $ITRM$-decidable. 2013 Carl, Merlin eng

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