# Church's thesis treated in two calcules with new calculation paradigm

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. This is partially overcome by a new calculation pardigm. Everything is done within FUME, the precise extension of predicate logic to a two-tier logic system of object-language and metalanguage.

1. The relatively old publication of 2006 is updated and gives the concrete calcule NU of fagative arithmetic. It follows the traditional way to define primitive recursive functions by means of a storage-based computer. However, instead of a register machine, the FAGATOR,  a  much simpler calculator is employed, that has only the commands succeed, repeat, delete and copy. The programs are coded by decimal numbers. The Spark-function is introduced as a challenge to the defenders of Church's thesis.

2. The Snark publication treats Church's thesis in the framework of concrete calcule LAMBDA of pinitive arithmetic. It uses an immediate definition of primitive recursive functions without programs based on register machines or similar storage-based computer, but rather on the PINITOR a new kind of calculator. An astonishing fact is it that this leads immediately to the so-called processive function, that contain the Ackermann function and more. Recursive functions can be represented as well. The richness of functions and the clear metalingual treatment opens the path to new insights. The primitive functions are coded by a decimal number. The Snark-function is introduced as a challenge to the defenders of Church's thesis.

3.  For a deeper understanding it is necessary to go into the details of concrete calcule LAMBDA of pinitive arithmetic. A long list of programming pinon codes is given, both on a first level and an advanced level, up to generator programming that eventually leads to processive functions. Furthermore the low values for both the Boojum-function and the Snark-function are listed, including some discussion of these calculative functions that are put forward as presumable counterexample for Church's thesis.

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