Abstract
In this article we provide an intrinsic characterization of the famous Howard–Bachmann ordinal in terms of a natural well-partial-ordering by showing that this ordinal can be realized as a maximal order type of a class of generalized trees with respect to a homeomorphic embeddability relation. We use our calculations to draw some conclusions about some corresponding subsystems of second order arithmetic. All these subsystems deal with versions of light-face \(\varPi ^1_1\)-comprehension.
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Van der Meeren, J., Rathjen, M. & Weiermann, A. An order-theoretic characterization of the Howard–Bachmann-hierarchy. Arch. Math. Logic 56, 79–118 (2017). https://doi.org/10.1007/s00153-016-0515-6
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DOI: https://doi.org/10.1007/s00153-016-0515-6
Keywords
- Well-partial-orderings
- Kruskal’s theorem
- Howard–Bachmann number
- Ordinal notation systems
- Natural well-orderings
- Maximal order type
- Collapsing function
- Recursively defined trees
- Tree-embeddabilities
- Proof-theoretical ordinal
- Impredicative theory
- Independence results
- Minimal bad sequence