# Church's thesis is questioned by new calculation paradigm

**Summary**

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 **FAGACUS **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 **FAGATOR** 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 **fagative** 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** fagative functions are recursive and to put forward a recursive function that is not fagative. A construction with **three-tier-multi-level-arbation** that includes a diagonalisation leads to the extravagant yet calculable **Spark-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** Gödel's first incompleteness sentence is in limbo.

**Download: ****Church's thesis ... **

32 pages, version 2.0 , first publication for priority 07.10.2006 arXiv cs/0610038

Although the publication has received some facelifting by version 2.0 it is somewhat outdated. The Spark counterexample is not described as thoroughly as the Snark counterexample on the next page