Gandy ordinal: Difference between revisions

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<ref>R. Gostanian. The Next Admissible Ordinal. Ann. Math. Logic, 17:171–203, 1979</ref>, which is greater than the least \(\alpha\) which is \(\alpha^+\)-stable, and less than the least \(\alpha\) which is \(\alpha^++1\)-stable.<ref>The Order of Reflection, J. P. Aguilera</ref> 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.