Ein Bezeichnungssystem für Ordinalzahlen

Ordinal arithmetic with simultaneously defined theta-functions

“…. , l. Note that the existence of such γ i ∈C m α ,θ m (α ) ∩ dom(θ m +1 ) ∩ α is guaranteed by (11) and Lemma 2.6. We put γ i := rt n m +1 (γ i ), Γ i := ϑ n m +1 (γ i ) for 1 ≤ i ≤ l, and Γ := Γ 1 + · · · + Γ l .…”

“…Note that Ψ > 0 since α ≥ ε Ωm + 1 +1 . Since α ∈ dom(θ m ) we have (11) α ∈C m (α ,θ m (α )), and by Lemma 2.16 we obtain Ψ ∈ dom(θ m +1 ). Set Ψ := rt n m +1 (Ψ ) and let γ i be such that Γ i =θ m +1 (γ i ) for i = 1, .…”

“…A somewhat related investigation concerning the binary function θ and its relation to Pfeiffer's system Σ, [11], has been carried out by Buchholz and Schütte in [5]. There an isomorphism between the systems θ(ω) and Σ is provided by a purely syntactical transformation on ordinal terms.…”