Die Beziehungen Zwischen den OrdinalzahlsystemenΣ Und $$\bar \Theta \left( \omega \right)$$

Abstract

In [9] wurde ein wohlgeordnetes System 27(N) von Ordinaltermen f'fir jede natiirliche Zahl N eingeffihrt. Die Ordinalterme der Systeme X(N) repr~isentieren nach H. Levitz

Ordinal arithmetic with simultaneously defined theta-functions

“…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.…”

“…A precise translation between ϑ, ϑ and the term systems used in [5] and other ordinal notations like those based on Buchholz' ψ-functions, see [4], or Gordeev-style axioms, see [8], seems rewarding and we leave this for future research. (Partial results based on embeddings have already been established.…”

Provable wellorderings of formal theories for transfinitely iterated inductive definitions

“…From (III) one derives in the well-known way (due to Gentzen 3 , transfinite induction up to any ordinal less than 0e nv +iO is provable in IDiv, which we will abbreviate by IDJvl-TI[< © e n^i 0 ] .…”

“…Each element of 5£ has the shape 0, a + b, ®ab or gab with 3 Cf. [3]. Note that the system £(JV) in [3] is a slight modification of that in [14].…”

“…[3]. Note that the system £(JV) in [3] is a slight modification of that in [14]. In [3] the first element of 2(N) is 0 instead of 1.…”