Abstract: "Map theory (MT) is a foundation of mathematics which is based on [lambda]-calculus instead of logic and has at least the same expressive power as ZFC set theory. This paper presents 'stable map theory' which is much easier to learn, teach and comprehend than 'original...

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Bibliographische Detailangaben

1. Verfasser:	Grue, Klaus (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	København 1996
Schriftenreihe:	Datalogisk Institut <København>: DIKU-Rapport 1996,10
Schlagworte:	Lambda calculus Mappings (Mathematics)
Zusammenfassung:	Abstract: "Map theory (MT) is a foundation of mathematics which is based on [lambda]-calculus instead of logic and has at least the same expressive power as ZFC set theory. This paper presents 'stable map theory' which is much easier to learn, teach and comprehend than 'original map theory' from 1992. Pedagogical simplicity is important since MT is a candidate for a common foundation of classical mathematics and computer science. MT has the benefit that it allows a complete integration of classical mathematics and computer science. As a particular example, the free mixing of quantification and general recursion has many applications. The long list of well-foundedness axioms in the original version (which corresponds to the list of proper axioms of ZFC) has been replaced by a single definition of 'classicality' in the stable version. Furthermore, the stable version has been enhanced by axioms of stability, minimality of fixed points and a particular kind of extensionality. The stability axiom represents a change of semantics compared to original MT which is why the new version is called 'stable'. The axioms of minimality of fixed points is particularly important since it allows to prove the axiom of transfinite induction in the original version. This paper presents stable MT and develops original MT in it. As a corollary of this development, also ZFC can be developed in stable MT. In addition, the paper presents an extension of the stable version which resembles NBG but is more general. The paper conjectures two specific structures to be models of stable MT. The stable version sheds some light on why it is consistent to allow infinite descending chains of the membership relation in a ZF-style system (as Aczel did in his theory of non-well-founded sets). This is because the Burali-Forti paradox is avoided by 'limitation' rather than 'well- foundedness' as made explicit in the definition of 'classical mathematical object'."
Beschreibung:	71 S.