Ptykes in GödelsT und Definierbarkeit von Ordinalzahlen

Summary

We prove two of the inequalities needed to obtain the following result on the ordinal values of ptykes of type 2, which are definable in Gödel'sT. LetG be a dilator satisfyingG(0)=ω, ∀x:G(x)≧x, and ∀η<Ω:G(η)<Ω, and letg be the ordinal function induced byG. Then sup{A(G)∣A ptyx of type 2 definable in Gödel'sT} = sup{x∣x is∑ ^g₁ -definable without parameters provably in KP(G)} =J ^g_{(2 +Id) (ω)} (0) = the “Bachmann-Howard ordinal relative tog”. KP(G) is obtained from Kripke-Platek set theory without urelements KP by adjoining a two-place relation symbolG and axioms expressing thatG is the graph of a total function from ordinals to ordinals.J ^g _D is an iteration hierarchy defined relative tog by primitive recursion on dilators. (2 +Id)_(ω) is the dilator$\mathop {\sup }\limits_{n< \omega } (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{2} + Id)_{(n)}$ with (2 +Id)₍₀₎≔1 and$(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{2} + Id)_{(n + 1)} : = (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{2} + Id)^{(2 + Id)_{(n)} }$. The “Bachmann-Howard ordinal relative tog” is the closure ordinal of a Bachmann hierarchy of lengthε _{Ω + 1}, which is built on an iteration ofg as initial function.

For the caseG=(1+Id)·ω, KP(G) is equivalent to Jäger's theory KPu, and the “Bachmann-Howard ordinal relative tog” is the usual “Bachmann-Howard ordinal”. For the caseG=Ξ₁ KP(G) can be replaced by Jäger's theory KPi, andg can be replaced by the functionλx. x ⁺.x ⁺ is the successor admissible ofx, andΞ ₁ is the sum of all recursive dilators in an arbitrary enumeration.