  In this work, we aim at understanding incompleteness in an abstract way via
metamathematical properties of formal theories. We systematically examine the
relationships between the following twelve important metamathematical
properties of arithmetical theories: Rosser, EI (Effectively inseparable), RI
(Recursively inseparable), TP (Turing persistent), EHU (essentially
hereditarily undecidable), EU (essentially undecidable), Creative,
$\mathbf{0}^{\prime}$ (theories with Turing degree $\mathbf{0}^{\prime}$), REW
(all RE sets are weakly representable), RFD (all recursive functions are
definable), RSS (all recursive sets are strongly representable), RSW (all
recursive sets are weakly representable). Given any two properties $P$ and $Q$
of these properties, we examine whether the property $P$ implies $Q$.
