Second order theories with ordinals and elementary comprehension

Some theories with positive induction of ordinal strength φω0

“…Let us finish this section by mentioning that it is also possible to provide a direct wellordering proof up to each ordinal less than ipcoO within IDf. For a similar argument the reader is referred to Jager and Strahm [18]. §3.…”

Notes on some second-order systems of iterated inductive definitions and Π11-comprehensions and relevant subsystems of set theory

“…For exactly the same reason, the statement (211) of [12, p. 328] is false and thus the proof of Theorem 3.4.6.15 of [12, p. 328] is flawed as well, which states W-KPl ≤ ψ Ω (Ω ω · ε 0 ), although this statement itself is true as well. 10 We will amend the proof of these two theorems in Section 9.3. As for (NUID 2 ν ) 0 , the third error of [12] lies in the very analysis of KPl r ν , as we also mentioned in Section 1.…”

“…In fact, the claim (5) is false and, as we will show in Sections 8,9, the proof-theoretic ordinal of KPl r ν is ψ Ω (Ω ν · ω). 10 In addition to the aforementioned crucial problem, the suggested application of Collapsing Theorem [12,Theorem 3.4.5.3] illustrated in Pohlers's proof of the claim (211) of [12, p. 328] is not generally valid. It is suggested there to infer…”

“…In particular, I am heavily indebted to Prof. Strahm; I realized one of the errors of [12] from his comment on my [8] and this gave me an initial motivation to write this paper. I thank Prof. Jäger and Prof. Strahm also for informing me of their [10] after my presentation on an early draft of the present paper in Münchenwiler. Finally, I very much thank the anonymous referee for his/her helpful corrections and suggestions.…”

The Universal Set and Diagonalization in Frege Structures

“…The applicative theory TON. The theory TON (total theory of operations and numbers), introduced and studied by Jäger & Strahm (1995), is the total version of the theory BON (basic theory of operations and numbers) introduced by Feferman & Jäger (1993) (see also the textbooks Beeson, 1985 andTroelstra &van Dalen, 1988 which contain presentations of first-order applicative theories). It is formulated in L t , the firstorder language of operations and numbers, comprising of individual variables x, y, z, v, w, f , i, m, n, .…”