Upper bounds for metapredicative Mahlo in explicit mathematics and admissible set theory

Abstract

Abstract. In this article we introduce systems for metapredicative Mahlo in explicit mathematics and admissible set theory. The exact upper proof-theoretic bounds of these systems are established. §1. Introduction. In classical set theory an ordinal K is called a Mahlo ordinal if it is a regular cardinal and if, for every normal function / from K to K, there exists a regular cardinal fi less than K so that {/(£) : £ < ju} C [i. The statement that there exists a Mahlo ordinal is a powerful set existence axiom going beyond theories like ZFC. It also outgrows the existence of inaccessible cardinals, hyper inaccessibles, hyperhyperinaccessible and the like.There is also an obvious recursive analogue of Mahlo ordinal. Typically, an ordinal a is baptized recursively Mahlo, if it is admissible and reflects every TI2 sentence on a smaller admissible ordinal. The corresponding formal theory KPM has been proof-theoretically analyzed by Rathjen [14, 15]. KPM is a highly impredicative theory, and its proof-theoretic strength is significantly beyond that of KPi, the second order theory (A 2 -CA) + (Bl) and Feferman's theory To, which are all proof-theoretically equivalent (cf. [3,6,10]).This article can be seen as a further contribution to the general program of metapredicativity. We have studied other metapredicative theories in Jager, Kahle, Setzer and Strahm [8], Jager and Strahm [11], and Strahm [21, 20]; there also some further background material can be found.One aim here is to look at metapredicative Mahlo in admissible set theory. The corresponding theory, named KPm°, is admissible set theory above the natural numbers as urelements plus n2 reflection on the admissibles. As induction principles we have complete induction on the natural numbers for sets, but do not include e induction.A further aim of this paper is to introduce the concept of Mahloness into explicit mathematics and to analyze the proof-theoretic strength of its metapredicative version. An extension of Feferman's theory T 0 by Mahlo axioms is studied in Jager and Studer [12]. Setzer [18] presents a related formulation in the framework of Martin-L6f type theory.For the formalization of Mahlo in explicit mathematics we work over the basic theory EETJ which comprises the axioms of applicative theories and has type

Variation on a theme of Schütte

“…(6) From assertions (4) and (6) we conclude s is smaller than t in the sense of ≺ so that the formula s ≺ t belongs to F. In addition, an inversion applied to (2) yields…”

“…For the binary Veblen function, providing a notation system for all ordinals less than the famous Feferman-Schütte ordinal Γ 0 , this is done, for example, in Schütte [11] and Pohlers [7] in full detail. Jäger and Strahm [6] and Strahm [14] employ the ternary Veblen function ϕ in connetion with their proof-theoretic analysis of metapredicative subsystems of explicit mathematics and set theory.…”

“…The ordinal ϕω00 mentioned below is the metapredicative Mahlo number; see Jäger and Strahm [6] for details.…”

“…It is proof-theoretic folklore that ϕε 0 0 is the proof-theoretic ordinal of the theory ∆ 3. Now we turn to a subsystem of set theory and consider the theory KPm 0 of Jäger and Strahm [6] . It formalizes that we have a Mahlo universe, i. e. Π 2 reflection on the admissibles; ∈-induction, however, is not available.…”

“…The language L(KPm 0 ) of KPm 0 contains the language L 2 modulo an obvious translation A * of any L 2 formula A. According to [6], KPm 0 is ϕω00-equivalent to PA ∞ and ϕω00 is the proof-theoretic ordinal of KPm 0 . As in the case above we can therefore deduce with the help of Theorem 8 that KPm 0 + ¬I * < (s) ϕω00 PA ∞ and |KPm 0 + ¬I * < (s)| = ϕω00, provided that, as before, s is a closed number term whose order type with respect to the well-ordering < is greater than or equal to ϕω00.…”