Contentual and Formal Aspects of Gentzen’s Consistency Proofs

ASSO, an innovative conceptual methodology which combines features of database design with the formal method B, has been defined in order to ensure the flexibility of semantic data models, the efficiency of object models and design correctness. Starting from a directed acyclic graph of classes supported by semantic data models, a formal mapping generates classes supported by object models. The classes supported by semantic data models are then extended with aspects of behavioural modelling: a relationship with the B model is established and the consistency proofs of the whole schema are reduced to small obligations of B. This chapter evidences how ASSO is based on model transformations. These have been introduced with various purposes: to map semantic data models to object models, to integrate static and dynamic modelling, to link formal and informal notations and to relate the conceptual schema and the logical schema of the methodology.

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Reading Gentzen's Three Consistency Proofs Uniformly

Journal of the Japan Association for Philosophy of Science ◽

10.4288/kisoron.41.1_1 ◽

2013 ◽

Vol 41 (1) ◽

pp. 1-22 ◽

Cited By ~ 1

Author(s):

Ryota AKIYOSHI ◽

Yuta TAKAHASHI

Keyword(s):

Consistency Proofs

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Certain predicates defined by induction schemata

Journal of Symbolic Logic ◽

10.2307/2266327 ◽

1953 ◽

Vol 18 (1) ◽

pp. 49-59 ◽

Cited By ~ 3

Author(s):

Hao Wang

Keyword(s):

Number Theory ◽

Set Theory ◽

Formal System ◽

General Notion ◽

Constant Number ◽

Natural Numbers ◽

System L ◽

Truth Definition ◽

Consistency Proofs ◽

Special Case

It is known that we can introduce in number theory (for example, the system Z of Hilbert-Bernays) by induction schemata certain predicates of natural numbers which cannot be expressed explicitly within the framework of number theory. The question arises how we can define these predicates in some richer system, without employing induction schemata. In this paper a general notion of definability by induction (relative to number theory), which seems to apply to all the known predicates of this kind, is introduced; and it is proved that in a system L1 which forms an extension of number theory all predicates which are definable by induction (hereafter to be abbreviated d.i.) according to the definition are explicitly expressible.In order to define such predicates and prove theorems answering to their induction schemata, we have to allow certain impredicative classes in L1. However, if we want merely to prove that for each constant number the special case of the induction schema for a predicate d.i. is provable, we do not have to assume the existence of impredicative classes. A certain weaker system L2, in which only predicative classes of natural numbers are allowed, is sufficient for the purpose. It is noted that a truth definition for number theory can be obtained in L2. Consistency proofs for number theory do not seem to be formalizable in L2, although they can, it is observed, be formalized in L1.In general, given any ordinary formal system (say Zermelo set theory), it is possible to define by induction schemata, in the same manner as in number theory, certain predicates which are not explicitly definable in the system. Here again, by extending the system in an analogous fashion, these predicates become expressible in the resulting system. The crucial predicate instrumental to obtaining a truth definition for a given system is taken as an example.

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