Bad
From Cantor's Attic
An ordinal is called bad [1] or non-Gandy[2] if the supremum of order types of $\alpha$-recursive well-orderings is less than the next admissible after $\alpha$. Since the least bad ordinal is quite large (larger than the least $\Pi_1^1$-reflecting ordinal [1]), this behavior appears unexpectedly when working with ordinals in the range of weakened versions of stability. [2] The least bad ordinal coincides with the least $\Sigma^1_1$-reflecting ordinal. [2]
The least bad ordinal is greater than the least ordinal that is \(\Pi^{1,set}_1\)-reflecting on the class of \(\Pi^{1,set}_1\)-reflecting ordinals. (J. P. Aguilera, The Order of Reflection (2019, arxiv preprint). Accessed 7 September 2022.)
Relation to other conditions
- $\Sigma_1^1$-reflection implies badness [3]
References
- Gostanian, R. The Next Admissible Ordinal. Annals of Mathematical Logic , 1979. www DOI bibtex
- Madore, David. A zoo of ordinals. , 2017. www bibtex