Authors: Hannes Hutzelmeyer
In connection with his so-called incompleteness theorem Gödel discovered the beta-function. The beta-function theorem is important for the representation of recursive functions in the concrete calcule ALPHA of Robinson arithmetic. The other features are composition and minimization of primitive recursive functions. Recursive functions are no part of Robinson arithmetic, but they are representable by certain formulae. The author has developed an approach to logics that comprises, but goes beyond predicate logic. The FUME method contains two tiers of precise languages: object-language Funcish and metalanguage Mencish. It allows for a very wide application in mathematics from recursion theory and axiomatic set theory with first-order logic, to higher-order logic theory of real numbers and so on. The concrete calcule LAMBDA of a natural number arithmetic with first-order logic has been defined by the author. It includes straight recursion and composition of functions, it contains a wide range of so-called compinitive functions, with processive functions far beyond primitive recursive functions. They include e.g. Ackermann's function and similar constructions. All recursive functions (that are obtained by minimization too) can be represented in LAMBDA . As long as there is no proof that all processive functions are minimitive recursive (recursive but not primitive recursive) one has the problem of representing them in concrete calcule ALPHA of Robinson arithmetic. As long as the challenge of such a proof is not met there is the conjecture that there are calculative functions that are not representable in Robinson arithmetic. An abstract calcule alphakappa of Robinson-Crusoe arithmetic shows that there exists an even weaker adequate arithmetic than Robinson's.
Comments: 10 Pages.
[v1] 2019-05-31 03:08:42
Unique-IP document downloads: 8 times
Vixra.org is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary. In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution. Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.
Add your own feedback and questions here:
You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.