Bad

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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

References

  1. Gostanian, R. The Next Admissible Ordinal. Annals of Mathematical Logic , 1979. www   DOI   bibtex
  2. Madore, David. A zoo of ordinals. , 2017. www   bibtex
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