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  1. Gerhard Jäger & Rico Zumbrunnen (forthcoming). Explicit mathematics and operational set theory: Some ontological comparisons. Association for Symbolic Logic: The Bulletin of Symbolic Logic.
    We discuss several ontological properties of explicit mathematics and operational set theory: global choice, decidable classes, totality and extensionality of operations, function spaces, class and set formation via formulas that contain the definedness predicate and applications.
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  2. Gerhard Jäger (2013). Operational Closure and Stability. Annals of Pure and Applied Logic 164 (7-8):813-821.
    In this article we introduce and study the notion of operational closure: a transitive set d is called operationally closed iff it contains all constants of OST and any operation f∈d applied to an element a∈d yields an element fa∈d, provided that f applied to a has a value at all. We will show that there is a direct relationship between operational closure and stability in the sense that operationally closed sets behave like Σ1 substructures of the universe. This leads (...)
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  3. Gerhard Jäger (2013). Rationalizable Signaling. Erkenntnis:1-34.
    An important finding of the game theoretic research on signaling games is the insight that under many circumstances, a signal obtains credibility by incurring costs to the sender. Therefore it seems questionable whether or not cheap talk—signals that are not payoff relevant—can serve to transmit information among rational agents. This issue is non-trivial in strategic interactions where the preferences of the players are not aligned. Researchers like Crawford & Sobel, Rabin, and Farrell demonstrated, however, that even in the case of (...)
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  4. Michael Franke & Gerhard Jäger (2012). Bidirectional Optimization From Reasoning and Learning in Games. Journal of Logic, Language and Information 21 (1):117-139.
    We reopen the investigation into the formal and conceptual relationship between bidirectional optimality theory (Blutner in J Semant 15(2):115–162, 1998 , J Semant 17(3):189–216, 2000 ) and game theory. Unlike a likeminded previous endeavor by Dekker and van Rooij (J Semant 17:217–242, 2000 ), we consider signaling games not strategic games, and seek to ground bidirectional optimization once in a model of rational step-by-step reasoning and once in a model of reinforcement learning. We give sufficient conditions for equivalence of bidirectional (...)
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  5. Gerhard Jäger & Dieter Probst (2011). The Suslin Operator in Applicative Theories: Its Proof-Theoretic Analysis Via Ordinal Theories. Annals of Pure and Applied Logic 162 (8):647-660.
    The Suslin operator is a type-2 functional testing for the well-foundedness of binary relations on the natural numbers. In the context of applicative theories, its proof-theoretic strength has been analyzed in Jäger and Strahm [18]. This article provides a more direct approach to the computation of the upper bounds in question. Several theories featuring the Suslin operator are embedded into ordinal theories tailored for dealing with non-monotone inductive definitions that enable a smooth definition of the application relation.
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  6. Gerhard Jäger & Thomas Studer (2011). A Buchholz Rule for Modal Fixed Point Logics. Logica Universalis 5 (1):1-19.
    Buchholz’s Ω μ+1-rules provide a major tool for the proof-theoretic analysis of arithmetical inductive definitions. The aim of this paper is to put this approach into the new context of modal fixed point logic. We introduce a deductive system based on an Ω-rule tailored for modal fixed point logic and develop the basic techniques for establishing soundness and completeness of the corresponding system. In the concluding section we prove a cut elimination and collapsing result similar to that of Buchholz (Iterated (...)
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  7. Gerhard Jäger (2009). Full Operational Set Theory with Unbounded Existential Quantification and Power Set. Annals of Pure and Applied Logic 160 (1):33-52.
    We study the extension of Feferman’s operational set theory provided by adding operational versions of unbounded existential quantification and power set and determine its proof-theoretic strength in terms of a suitable theory of sets and classes.
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  8. Gerhard Jäger (2007). On Feferman's Operational Set Theory. Annals of Pure and Applied Logic 150 (1):19-39.
    We study and some of its most important extensions primarily from a proof-theoretic perspective, determine their consistency strengths by exhibiting equivalent systems in the realm of traditional set theory and introduce a new and interesting extension of which is conservative over.
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  9. Gerhard Jäger (2007). The Evolution of Convex Categories. Linguistics and Philosophy 30 (5):551-564.
    Gärdenfors (Conceptual spaces, 2000) argues that the semantic domains that natural language deals with have a geometrical structure. He gives evidence that simple natural language adjectives usually denote natural properties, where a natural property is a convex region of such a “conceptual space.” In this paper I will show that this feature of natural categories need not be stipulated as basic. In fact, it can be shown to be the result of evolutionary dynamics of communicative strategies under very general assumptions.
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  10. Gerhard Jäger & Robert van Rooij (2007). Language Structure: Psychological and Social Constraints. Synthese 159 (1):99 - 130.
    In this article we discuss the notion of a linguistic universal, and possible sources of such invariant properties of natural languages. In the first part, we explore the conceptual issues that arise. In the second part of the paper, we focus on the explanatory potential of horizontal evolution. We particularly focus on two case studies, concerning Zipf's Law and universal properties of color terms, respectively. We show how computer simulations can be employed to study the large scale, emergent, consequences of (...)
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  11. Gerhard Jäger & Robert van Rooij (2007). Language Structure: Psychological and Social Constraints. Synthese 159 (1):99 - 130.
    In this article we discuss the notion of a linguistic universal, and possible sources of such invariant properties of natural languages. In the first part, we explore the conceptual issues that arise. In the second part of the paper, we focus on the explanatory potential of horizontal evolution. We particularly focus on two case studies, concerning Zipf’s Law and universal properties of color terms, respectively. We show how computer simulations can be employed to study the large scale, emergent, consequences of (...)
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  12. Luca Alberucci & Gerhard Jäger (2005). About Cut Elimination for Logics of Common Knowledge. Annals of Pure and Applied Logic 133 (1):73-99.
    The notions of common knowledge or common belief play an important role in several areas of computer science , in philosophy, game theory, artificial intelligence, psychology and many other fields which deal with the interaction within a group of “agents”, agreement or coordinated actions. In the following we will present several deductive systems for common knowledge above epistemic logics –such as K, T, S4 and S5 –with a fixed number of agents. We focus on structural and proof-theoretic properties of these (...)
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  13. Gerhard Jäger & Dieter Probst (2005). Corrigendum to “Variation on a Theme of Schütte”. Mathematical Logic Quarterly 51 (6):642-642.
    We give a corrected definition for the paper [1] mentioned in the title.
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  14. Gerhard Jäger & Thomas Strahm (2005). Reflections on Reflections in Explicit Mathematics. Annals of Pure and Applied Logic 136 (1-2):116-133.
    We give a broad discussion of reflection principles in explicit mathematics, thereby addressing various kinds of universe existence principles. The proof-theoretic strength of the relevant systems of explicit mathematics is couched in terms of suitable extensions of Kripke–Platek set theory.
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  15. Gerhard Jäger (2004). An Intensional Fixed Point Theory Over First Order Arithmetic. Annals of Pure and Applied Logic 128 (1-3):197-213.
    The purpose of this article is to present a new theory for fixed points over arithmetic which allows the building up of fixed points in a very nested and entangled way. But in spite of its great expressive power we can show that the proof-theoretic strength of our theory—which is intensional in a meaning to be described below—is characterized by the Feferman–Schütte ordinal Γ0. Our approach is similar to the building up of fixed points over state spaces in the propositional (...)
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  16. Gerhard Jäger (2004). Relationale Grammatik. Journal of Logic, Language and Information 13 (4):521-525.
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  17. Gerhard Jäger (2004). Residuation, Structural Rules and Context Freeness. Journal of Logic, Language and Information 13 (1):47-59.
    The article presents proofs of the context freeness of a family of typelogical grammars, namely all grammars that are based on a uni- ormultimodal logic of pure residuation, possibly enriched with thestructural rules of Permutation and Expansion for binary modes.
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  18. Gerhard Jager & Structural Rules Residuation (2004). Alexander Koller, Ralph Debusmann, Malte Gabsdil, and Kristina Striegnitz/Put My Galakmid Coin Into the Dispenser and Kick It: Computational Linguistics and Theorem Proving in a Computer Game 187–206. Journal of Logic, Language and Information 13:537-539.
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  19. Gerhard Jäger (2003). Resource Sharing in Type Logical Grammar. In R. Oehrle & J. Kruijff (eds.), Resource Sensitivity, Binding, and Anaphora. Kluwer. 97--121.
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  20. Gerhard Jäger (2003). Towards an Explanation of Copula Effects. Linguistics and Philosophy 26 (5):557-593.
    This paper deals with a series of semantic contrasts between the copula be and the preposition as, two functional elements that both head elementary predication structures. It will be argued that the meaning of as is a type lowering device shifting the meaning of its complement NP from the type of generalized quantifiers to the type of properties (where properties are conceived as relations between individuals and situations), while the copula be induces a type coercion from (partial) situations to (total) (...)
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  21. Gerhard Jäger (2002). Some Notes on the Formal Properties of Bidirectional Optimality Theory. Journal of Logic, Language and Information 11 (4):427-451.
    In this paper, we discuss some formal properties of the model ofbidirectional Optimality Theory that was developed inBlutner (2000). We investigate the conditions under whichbidirectional optimization is a well-defined notion, and we give aconceptually simpler reformulation of Blutner's definition. In thesecond part of the paper, we show that bidirectional optimization can bemodeled by means of finite state techniques. There we rely heavily onthe related work of Frank and Satta (1998) about unidirectionaloptimization.
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  22. Gerhard Jäger & Thomas Studer (2002). Extending the System of Explicit Mathematics: The Limit and Mahlo Axioms. Annals of Pure and Applied Logic 114 (1-3):79-101.
    In this paper we discuss extensions of Feferman's theory T 0 for explicit mathematics by the so-called limit and Mahlo axioms and present a novel approach to constructing natural recursion-theoretic models for systems of explicit mathematics which is based on nonmonotone inductive definitions.
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  23. Gerhard Jäger (2001). First Order Theories for Nonmonotone Inductive Definitions: Recursively Inaccessible and Mahlo. Journal of Symbolic Logic 66 (3):1073-1089.
    In this paper first order theories for nonmonotone inductive definitions are introduced, and a proof-theoretic analysis for such theories based on combined operator forms a la Richter with recursively inaccessible and Mahlo closure ordinals is given.
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  24. Gerhard Jäger, Reinhard Kahle & Thomas Studer (2001). Universes in Explicit Mathematics. Annals of Pure and Applied Logic 109 (3):141-162.
    This paper deals with universes in explicit mathematics. After introducing some basic definitions, the limit axiom and possible ordering principles for universes are discussed. Later, we turn to least universes, strictness and name induction. Special emphasis is put on theories for explicit mathematics with universes which are proof-theoretically equivalent to Feferman's.
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  25. Gerhard Jäger & Thomas Strahm (2001). Upper Bounds for Metapredicative Mahlo in Explicit Mathematics and Admissible Set Theory. Journal of Symbolic Logic 66 (2):935-958.
    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.
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  26. Gerhard Jäger & Thomas Strahm (2000). Fixed Point Theories and Dependent Choice. Archive for Mathematical Logic 39 (7):493-508.
    In this paper we establish the proof-theoretic equivalence of (i) $\hbox {\sf ATR}$ and $\widehat{\hbox{\sf ID}}_{\omega}$ , (ii) $\hbox{\sf ATR}_0+ (\Sigma^1_1-\hbox{\sf DC})$ and $\widehat{\hbox {\sf ID}}_{<\omega^\omega} , and (iii) $\hbox {\sf ATR}+(\Sigma^1_1-\hbox{\sf DC})$ and $\widehat{\hbox {\sf ID}}_{<\varepsilon_0} $.
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  27. Gerhard Jäger, Reinhard Kahle, Anton Setzer & Thomas Strahm (1999). The Proof-Theoretic Analysis of Transfinitely Iterated Fixed Point Theories. Journal of Symbolic Logic 64 (1):53-67.
    This article provides the proof-theoretic analysis of the transfinitely iterated fixed point theories $\widehat{ID}_\alpha and \widehat{ID}_{ the exact proof-theoretic ordinals of these systems are presented.
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  28. Gerhard Jäger & Thomas Strahm (1999). Bar Induction and Ω Model Reflection. Annals of Pure and Applied Logic 97 (1-3):221-230.
    We show that the principle of ω model reflection for Π1n − 1 formulas is equivalent over ACA0 to the scheme of Π1n bar induction. This extends and refines previous results of Friedman and Simpson.
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  29. Gerhard Jäger & Robert F. Stärk (1998). A Proof-Theoretic Framework for Logic Programming. In Samuel R. Buss (ed.), Handbook of Proof Theory. Elsevier. 639--682.
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  30. Gerhard Jäger (1997). Power Types in Explicit Mathematics? Journal of Symbolic Logic 62 (4):1142-1146.
    In this note it is shown that in explicit mathematics the strong power type axiom is inconsistent with (uniform) elementary comprehension and discuss some general aspects of power types in explicit mathematics.
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  31. Solomon Feferman & Gerhard Jäger (1996). Systems of Explicit Mathematics with Non-Constructive Μ-Operator. Part II. Annals of Pure and Applied Logic 79 (1):37-52.
    This paper is mainly concerned with proof-theoretic analysis of some second-order systems of explicit mathematics with a non-constructive minimum operator. By introducing axioms for variable types we extend our first-order theory BON to the elementary explicit type theory EET and add several forms of induction as well as axioms for μ. The principal results then state: EET plus set induction is proof-theoretically equivalent to Peano arithmetic PA <0).
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  32. Gerhard Jager & Thomas Strahm (1996). Some Theories with Positive Induction of Ordinal Strength $Varphiomega 0$. Journal of Symbolic Logic 61 (3):818-842.
    This paper deals with: (i) the theory $\mathrm{ID}^{\tt\#}_1$ which results from $\widehat{\mathrm{ID}}_1$ by restricting induction on the natural numbers to formulas which are positive in the fixed point constants, (ii) the theory $\mathrm{BON}(\mu)$ plus various forms of positive induction, and (iii) a subtheory of Peano arithmetic with ordinals in which induction on the natural numbers is restricted to formulas which are $\Sigma$ in the ordinals. We show that these systems have proof-theoretic strength $\varphi\omega 0$.
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  33. Gerhard Jäger & Thomas Strahm (1996). Some Theories with Positive Induction of Ordinal Strength Φω. Journal of Symbolic Logic 61 (3):818-842.
    This paper deals with: (i) the theory ID # 1 which results from $\widehat{\mathrm{ID}}_1$ by restricting induction on the natural numbers to formulas which are positive in the fixed point constants, (ii) the theory BON(μ) plus various forms of positive induction, and (iii) a subtheory of Peano arithmetic with ordinals in which induction on the natural numbers is restricted to formulas which are Σ in the ordinals. We show that these systems have proof-theoretic strength φω 0.
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  34. Sergei Artemov, George Boolos, Erwin Engeler, Solomon Feferman, Gerhard Jäger & Albert Visser (1995). Preface. Annals of Pure and Applied Logic 75 (1-2):1.
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  35. Gerhard Jäger & Thomas Strahm (1995). Second Order Theories with Ordinals and Elementary Comprehension. Archive for Mathematical Logic 34 (6):345-375.
    We study elementary second order extensions of the theoryID 1 of non-iterated inductive definitions and the theoryPA Ω of Peano arithmetic with ordinals. We determine the exact proof-theoretic strength of those extensions and their natural subsystems, and we relate them to subsystems of analysis with arithmetic comprehension plusΠ 1 1 comprehension and bar induction without set parameters.
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  36. Gerhard Jäger & Thomas Strahm (1995). Totality in Applicative Theories. Annals of Pure and Applied Logic 74 (2):105-120.
    In this paper we study applicative theories of operations and numbers with the non-constructive minimum operator in the context of a total application operation. We determine the proof-theoretic strength of such theories by relating them to well-known systems like Peano Arithmetic PA and the system <0 of second order arithmetic. Essential use will be made of so-called fixed-point theories with ordinals, certain infinitary term models and Church-Rosser properties.
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  37. Brigitte Hosli & Gerhard Jager (1994). About Some Symmetries of Negation. Journal of Symbolic Logic 59 (2):473 - 485.
    This paper deals with some structural properties of the sequent calculus and describes strong symmetries between cut-free derivations and derivations, which do not make use of identity axioms. Both of them are discussed from a semantic and syntactic point of view. Identity axioms and cuts are closely related to the treatment of negation in the sequent calculus, so the results of this article explain some nice symmetries of negation.
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  38. Brigitte Hösli & Gerhard Jäger (1994). About Some Symmetries of Negation. Journal of Symbolic Logic 59 (2):473-485.
    This paper deals with some structural properties of the sequent calculus and describes strong symmetries between cut-free derivations and derivations, which do not make use of identity axioms. Both of them are discussed from a semantic and syntactic point of view. Identity axioms and cuts are closely related to the treatment of negation in the sequent calculus, so the results of this article explain some nice symmetries of negation.
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  39. Solomon Feferman & Gerhard Jäger (1993). Systems of Explicit Mathematics with Non-Constructive Μ-Operator. Part I. Annals of Pure and Applied Logic 65 (3):243-263.
    Feferman, S. and G. Jäger, Systems of explicit mathematics with non-constructive μ-operator. Part I, Annals of Pure and Applied Logic 65 243-263. This paper is mainly concerned with the proof-theoretic analysis of systems of explicit mathematics with a non-constructive minimum operator. We start off from a basic theory BON of operators and numbers and add some principles of set and formula induction on the natural numbers as well as axioms for μ. The principal results then state: BON plus set induction (...)
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  40. Gerhard Jäger (1993). Fixed Points in Peano Arithmetic with Ordinals. Annals of Pure and Applied Logic 60 (2):119-132.
    Jäger, G., Fixed points in Peano arithmetic with ordinals, Annals of Pure and Applied Logic 60 119-132. This paper deals with some proof-theoretic aspects of fixed point theories over Peano arithmetic with ordinals. It studies three such theories which differ in the principles which are available for induction on the natural numbers and ordinals. The main result states that there is a natural theory in this framework which is a conservative extension of Peano arithmeti.
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  41. Gerhard Jager & Barbara Primo (1992). About the Proof-Theoretic Ordinals of Weak Fixed Point Theories. Journal of Symbolic Logic 57 (3):1108 - 1119.
    This paper presents several proof-theoretic results concerning weak fixed point theories over second order number theory with arithmetic comprehension and full or restricted induction on the natural numbers. It is also shown that there are natural second order theories which are proof-theoretically equivalent but have different proof-theoretic ordinals.
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  42. Gerhard Jäger & Barbara Primo (1992). About the Proof-Theoretic Ordinals of Weak Fixed Point Theories. Journal of Symbolic Logic 57 (3):1108-1119.
    This paper presents several proof-theoretic results concerning weak fixed point theories over second order number theory with arithmetic comprehension and full or restricted induction on the natural numbers. It is also shown that there are natural second order theories which are proof-theoretically equivalent but have different proof-theoretic ordinals.
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  43. Gerhard Jager (1991). Review: Wilfrid Buchholz, Kurt Schutte, Proof Theory of Impredicative Subsystems of Analysis. [REVIEW] Journal of Symbolic Logic 56 (1):332-333.
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  44. Gerhard Jäger (1991). Between Constructive Mathematics and PROLOG. Archive for Mathematical Logic 30 (5-6):297-310.
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  45. Gerhard Jäger (1991). Buchholz Wilfried and Schütte Kurt. Proof Theory of Impredicative Subsystems of Analysis. Studies in Proof Theory. Bibliopolis, Naples 1988, 122 Pp. [REVIEW] Journal of Symbolic Logic 56 (1):332-333.
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  46. Gerhard Jäger (1986). A Boundedness Theorem in mathrmID1 (W). Journal of Symbolic Logic 51 (4):942 - 947.
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  47. Gerhard Jäger (1986). Countable Admissible Ordinals and Dilators. Mathematical Logic Quarterly 32 (25‐30):451-456.
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  48. Gerhard Jäger (1986). Theories for Admissible Sets: A Unifying Approach to Proof Theory. Bibliopolis.
     
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  49. Gerhard Jäger (1984). A Version of Kripke‐Platek Set Theory Which is Conservative Over Peano Arithmetic. Mathematical Logic Quarterly 30 (1‐6):3-9.
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  50. Gerhard Jäger (1984). Ρ-Inaccessible Ordinals, Collapsing Functions and a Recursive Notation System. Archive for Mathematical Logic 24 (1):49-62.
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