  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.
