Gerhard J ager and Thomas Studer, Extending the system T 0 of explicit mathematics: the limit and Mahlo axioms, Annals of Pure and Applied Logic, to appear.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to names of types there exists a uniformly given universe that is closed under this operation. We note that EMA does not include inductive generation and that induction on the natural number is restricted to types. For more information concerning EMA plus inductive generation see Jager and Studer [12]. # Institut fur Informatik und angewandte Mathematik, Universitat Bern, Neubruckstrasse 10, CH 3012 Bern, Switzerland. Email: strahm iam.unibe.ch 1 The theory KPm 0 , on the other hand, is Rathjen s theory KPM (cf. Rathjen [17, 18] with induction on the natural numbers restricted to sets ....

J ager, G., and Studer, T. Extending the system T 0 of explicit mathematics: the limit and Mahlo axioms. Submitted.

....but do not include # induction. 1 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 Lof 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 existence axioms for elementary comprehension and join. ....

....systems. No methods of impredicative proof theory are used in our analysis of EMA and EMA (L I N ) so that the metapredicativity of both systems is established. Impredicative Mahlo in explicit mathematics is obtained by adding the principle of inductive generation to EMA, cf. Jager and Studer [12]. 4 The theory OMA In this section we introduce the ordinal theory OMA for the Mahlo axiom. It is a first order theory with ordinals tailored for dealing with certain nonmonotone inductive definitions which provides the appropriate framework for modelling our theory EMA. In the next section we ....

J ager, G., and Studer, T. Extending the system T 0 of explicit mathematics: the limit and Mahlo axioms. In preparation. 30

....has been given by Rathjen [43] making use of methods of traditional impredicative proof theory. The corresponding system FID( POS # 0 # , # 0 1 ] for first order inductive definitions treating combined operators from [POS # 0 # , # 0 1 ] was studied in Jager [23] and Jager and Studer [38]. B. Full and metapredicative explicit Mahlo. Full and metapredicative explicit Mahlo will be introduced in Section 6. Then we also state the respective results concerning their proof theoretic strength. In a nutshell: full explicit Mahlo is obtained from Feferman s T 0 by adding the Mahlo ....

....definability. Both systems provide a first step towards nonmonotone inductive definitions. This becomes very perspicuous in the context of the new model constructions for explicit mathematics given the help of certain classes of nonmonotone inductive definitions in Jager [23] and Jager and Studer [38] and Studer [60] Nonmonotone inductive definability is even more important if the Mahlo axioms (M1) and (M2) are added to T 0 ; call the resulting theory T 0 (M) for simplicity. In the next section we will try to convey an idea how nonmonotone inductive definitions can be used for modeling ....

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Gerhard J ager and Thomas Studer, Extending the system T 0 of explicit mathematics: the limit and Mahlo axioms, Annals of Pure and Applied Logic, to appear.

.... definition is needed to establish a model, whereas minimality is not necessary (cf. Marzetta [21] Formalizing this procedure in # ID 1 yields a model for EETJ (Tot) F I N ) More on inductive model constructions for systems of explicit mathematics can be found in Jager and Studer [20]. 13 5 Embedding # str into explicit mathematics In this section we are going to carry out the embedding of # str into the theory EETJ (Tot) F I N ) of explicit mathematics. First, we represent each pretype T of # str by a natural number T # , which will be called symbol for ....

Gerhard Jager and Thomas Studer. Extending the system T 0 of explicit mathematics: the limit and Mahlo axioms. Submitted. 23

....explicit mathematics can be found, for example, in Jager and Strahm [15] and Strahm [29] always in connection with theories of predicative or metapredicative strength. Universes are also crucial for dealing with Mahloness in explicit mathematics, as shown in the forthcoming paper Jager and Studer [16]. In Kahle [18] universes are studied for Frege structures, i.e. truth theories corresponding to explicit mathematics. The purpose of this article is to clarify several principle aspects of universes in explicit mathematics and to present them in compact form. After introducing some basic ....

....[11] a so called limit axiom can easily be added. By making use of the generator #, one assigns to each name x the name #x of a universe containing x, i.e. Lim) #x(#(x) # U(#x) # x # #x) The standard model constructions of Jager and Strahm [15] for metapredicative and Jager and Studer [16] for impredicative Mahlo provide natural models for (Lim) The proof theoretic strengths of (Lim) in the context of elementary comprehension and join plus type or formula induction on the natural numbers have been analyzed in Kahle [19] and Strahm [29] Although, in many situations, Lim) is ....

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J ager, G., and Studer, T. Extending the system T 0 of explicit mathematics: the limit and Mahlo axioms. Submitted.

.... de nition is needed to establish a model, whereas minimality is not necessary (cf. Marzetta [21] Formalizing this procedure in b ID 1 yields a model for EETJ (Tot) F I N ) More on inductive model constructions for systems of explicit mathematics can be found in J ager and Studer [20]. 13 5 Embedding fg str into explicit mathematics In this section we are going to carry out the embedding of fg str into the theory EETJ (Tot) F I N ) of explicit mathematics. First, we represent each pretype T of fg str by a natural number T , which will be called symbol ....

Gerhard Jager and Thomas Studer. Extending the system T 0 of explicit mathematics: the limit and Mahlo axioms. Submitted. 23

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