## Upper bounds for metapredicative Mahlo in explicit mathematics and admissible set theory (0)

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Venue: | Journal of Symbolic Logic |

Citations: | 20 - 14 self |

### BibTeX

@ARTICLE{Jäger_upperbounds,

author = {Gerhard Jäger and Thomas Strahm},

title = {Upper bounds for metapredicative Mahlo in explicit mathematics and admissible set theory},

journal = {Journal of Symbolic Logic},

year = {},

volume = {66},

pages = {935--958}

}

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### Abstract

In this article we introduce systems for metapredicative Mahlo in explicit mathematics and admissible set theory. The exact upper prooftheoretic bounds of these systems are established. 1 Introduction In classical set theory an ordinal # is called a Mahlo ordinal if it is a regular cardinal and if, for every normal function f from # to #, there exists a regular cardinal less than # so that {f(#) : # < } # . 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 # is baptized recursively Mahlo, if it is admissible and reflects every # 2 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-the...

### Citations

72 |
Foundations of constructive mathematics. Metamathematical studies., volume 6 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3
- Beeson
- 1985
(Show Context)
Citation Context ...ces of type variables U1, . . . , Um and individual terms s1, . . . , sn, respectively, whose length is given by the context. The logic of EMA is Beeson’s classical logic of partial terms (cf. Beeso=-=n [2] o-=-r Troelstra and Van Dalen [22]) for the individuals and classical logic with 7sequality for the types. Observe that Beeson’s formalization includes the usual strictness axioms. Before turning to our... |

67 |
Constnictive theories of functions and classes
- Feferman
- 1979
(Show Context)
Citation Context ...ive theory, and its proof-theoretic strength is significantly beyond that of KPi, the second order theory (∆ 1 2-CA) + (BI) and Feferman’s theory T0, 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 Jäger, Kahle, Setzer and Strahm [8], Jäger and Stra... |

50 |
Proof Theory. An Introduction
- Pohlers
(Show Context)
Citation Context ... ϕ function which we are now going to define. The usual Veblen hierarchy, generated by the binary function ϕ, starting off with the function ϕ0β = ω β , is well known from the literature, cf. Po=-=hlers [13] or Schütte [17-=-]. The ternary ϕ function is obtained as a straightforward generalization of the binary case by defining ϕαβγ inductively as follows: (i) ϕ0βγ is just ϕβγ; (ii) if α > 0, then ϕα0γ deno... |

48 | Systems of explicit mathematics with non-constructive µ-operator
- Feferman, Jäger
- 1993
(Show Context)
Citation Context ...zation but directly as an infinite axiom scheme. An L formula A is called elementary if it contains neither the relation symbol ℜ nor bound type variables. The following theorem of Feferman and Jäg=-=er [4]-=- shows that this scheme of uniform elementary comprehension is provable from our finite axiomatization. Join and uniqueness of generators are not needed for this argument. 9sTheorem 2 For every elemen... |

33 |
Eine beweistheoretische Untersuchung von (∆12-CA)+(BI) und verwandter Systeme
- Jäger, Pohlers
- 1982
(Show Context)
Citation Context ...ive theory, and its proof-theoretic strength is significantly beyond that of KPi, the second order theory (∆ 1 2-CA) + (BI) and Feferman’s theory T0, 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 Jäger, Kahle, Setzer and Strahm [8], Jäger and Stra... |

23 |
The proof-theoretic analysis of transfinitely iterated fixed point theories
- Jäger, Kahle, et al.
- 1999
(Show Context)
Citation Context ...ent (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 Jäger, Kahle, Setzer and Strahm=-= [8],-=- Jäger and Strahm [11], and Strahm [19, 21]; there also some further background material can be found. One aim here is to look at metapredicative Mahlo in admissible set theory. The corresponding the... |

13 | Extending the system T0 of explicit mathematics: the limit and Mahlo axioms
- Jäger, Studer
(Show Context)
Citation Context ... of Mahloness into explicit mathematics and to analyze the proof-theoretic strength of its metapredicative version. An extension of Feferman’s theory T0 by Mahlo axioms is studied in Jäger and Stud=-=er [12].-=- Setzer [18] presents a related formulation in the framework of Martin-Löf type theory. For the formalization of Mahlo in explicit mathematics we work over the basic theory EETJ which comprises the a... |

9 |
Separating fragments of bounded arithmetic
- Beckmann
- 1996
(Show Context)
Citation Context ... ∗ . From now on we take the liberty to also write TI(≺, A) for the corresponding formulas in the languages L and LO. It only remains to apply one of the usual boundedness theorems (cf. e.g. Beckm=-=ann [1] or Schütte [17]) s-=-tating that H α 0 TI(≺, Q) =⇒ | ≺ | ≤ ω · α, for all α and all primitive recursive wellorderings ≺; here | ≺ | is the ordertype of ≺ as usual. Together with the lower bound computat... |

8 |
Die Konstruktible Hierarchie als Hilfsmittel zur beweistheoretischen Untersuchung von Teilsystemen der Mengenlehre und Analysis
- Jäger
- 1979
(Show Context)
Citation Context ...rice for obtaining a metapredicative theory. The situation here is analogue to that for theories of iterated admissible sets dealing with recursive inaccessibility. The theory KPi introduced in Jäger=-= [5] c-=-an be considered as a formalized 4sapproach to a recursively inaccessible universe; it contains full induction on the natural numbers and full ∈ induction. KPi is fairly strong and prooftheoreticall... |

8 | Explicit Mathematics
- Feferman, Jäger, et al.
(Show Context)
Citation Context ...It is an interesting topic to see what kind of ordering principles for universes can be consistently added to the previous axioms. This question is discussed at full length in Jäger, Kahle and Studer=-= [9]-=-, and it is shown there that one must not be too liberal. As a consequence of these considerations we do not claim linearity and connectivity for arbitrary universes, but only for so-called normal uni... |

8 |
Fixed point theories and dependent choice. Archive for Mathematical Logic 39
- Jäger, Strahm
- 2000
(Show Context)
Citation Context ...This article can be seen as a further contribution to the general program of metapredicativity. We have studied other metapredicative theories in Jäger, Kahle, Setzer and Strahm [8], Jäger and Strah=-=m [11]-=-, and Strahm [19, 21]; 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 0 , is a... |

6 |
The strength of admissibility without foundation, The
- Jäger
- 1984
(Show Context)
Citation Context ...ystems of strength ϕε000. 2 The theory KPm 0 In this section we introduce the metapredicative theory KPm 0 for a recursively inaccessible Mahlo universe. Basically, KPm 0 is the theory KPi 0 of Jäg=-=er [7] aug-=-mented by an axiom scheme for Π2 reflection on the admissibles. It is equivalent to the theory KPM of Rathjen [14] if complete induction on ω is restricted to sets and all other forms of ∈ inducti... |

4 |
A well-ordering proof for Feferman’s theory T0. Archiv für mathematische Logik und Grundlagenforschung 23
- Jäger
- 1983
(Show Context)
Citation Context ...ive theory, and its proof-theoretic strength is significantly beyond that of KPi, the second order theory (∆ 1 2-CA) + (BI) and Feferman’s theory T0, 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 Jäger, Kahle, Setzer and Strahm [8], Jäger and Stra... |