Gandy ordinal
For an ordinal \(\alpha\), let \(\delta(\alpha)\) be the supremum of the order-types of \(\alpha\)-recursive well-orderings on a subset of \(\alpha\). An ordinal \(\alpha\) is called Gandy if \(\delta(\alpha)\) is equal to the next admissible ordinal after \(\alpha\), i.e. \(\delta(\alpha) = \alpha^+\). For example, \(\omega\) is trivially Gandy, and in general, any ordinal below the least recursively inaccessible ordinal is Gandy. An ordinal which is not Gandy is called non-Gandy or, colloquially, bad. The least bad ordinal is equal to the least ordinal which is \(\Sigma^1_1\)-reflecting[1], which is greater than the least \(\alpha\) which is \(\alpha^+\)-stable, and less than the least \(\alpha\) which is \(\alpha^++1\)-stable.[2] The structure of stability at the level of bad ordinals and beyond becomes a lot more complex, due to the highly nonlinear nature of iterated \(\Sigma^1_1\)- and \(\Pi^1_1\)-reflection, and one needing more and more iterations of \(\delta\) to reach the next admissible ordinal.
Proper usage of previously ignored, intricate bad ordinal structure is believed to be essential to maximising the strength of ordinal collapsing functions and associated ordinal notations. For example, it is believed that the least bad ordinal is a suitable ordinal to assign to the term \(\Omega\) in Dmytro Taranovsky's "Degrees of Reflection".[3]