Church's thesis is questioned by new calculation paradigm

TL;DR

This paper introduces a new computational paradigm that challenges Church's thesis by constructing a calculable function, the Snark-function, which is not arbative, thus questioning the universality of recursive functions.

Contribution

It proposes a multi-level arbation framework and a new computer model, ARBACUS, to demonstrate functions outside the scope of traditional recursive functions.

Findings

ARBACUS can compute all primitive recursive functions.

Multi-level arbation allows calculation of non-primitive recursive arbative functions.

The Snark-function is calculable but not arbative, challenging Church's thesis.

Abstract

Church's thesis claims that all effecticely calculable functions are recursive. A shortcoming of the various definitions of recursive functions lies in the fact that it is not a matter of a syntactical check to find out if an entity gives rise to a function. Eight new ideas for a precise setup of arithmetical logic and its metalanguage give the proper environment for the construction of a special computer, the ARBACUS computer. Computers do not come to a necessary halt; it is requested that calculators are constructed on the basis of computers in a way that they always come to a halt, then all calculations are effective. The ARBATOR is defined as a calculator with two-layer-computation. It allows for the calculation of all primitive recursive functions, but multi-level-arbation also allows for the calculation of other arbative functions that are not primitive recursive. The new paradigm of calculation does not have the above mentioned shortcoming. The defenders of Church's thesis are challenged to show that exotic arbative functions are recursive and to put forward a recursive function that is not arbative. A construction with three-tier-multi-level-arbation that includes a diagonalisation leads to the extravagant yet calculable Snark-function that is not arbative. As long as it is not shown that all exotic arbative functions and particularily the Snark-function are arithmetically representable Goedel's first incompleteness sentence is in limbo.