Some problems on natural well orderings
They come with a short definition. Make this precise!
They have intriguing algebraic
properties (functoriality, end-extendibility, relativization,...).
In particular they are analytic functors in the sense of Joyal following
Hazegawa. Collect these!
Their well-foundedness can be proved constructively.
They come with an additive logical limit law. Provide more examples!
They come with an multiplicative global logical limit law. Provide more examples!
They come with a robust hierarchy of fast growing functions.
: Provide more examples!
They come with intriguing slow growing hierarchies of
varying rate of growth: Provide more examples!.
They are maximal linear extensions of
natural well partial orders: Provide more examples!
They induce reduction orders stable under
substitution and application of function symbols: Provide more examples!
Beklemishev: They emerge from provability logic.
Arai (perhaps following Gentzen): They emerge from finite proof figures.
Ackermann: They emerge from epsilon substitution.
Multiplicative codings of ordinals below epsilon_0 have
slowly varying count functions: Provide more examples!
Multiplicative codings of ordinals above epsilon_0 have
regularly varying count functions: Provide more examples!
Additive codings of ordinals below epsilon_0 are in the
Compton class RT_0: Provide more examples!
Additive codings of ordinals above epsilon_0 are in the
Compton class RT_a for some a>0: Provide more examples!