Set Theory and Logic


Programming Primitive Recursive Functions and Beyond

Authors: Hannes Hutzelmeyer

In addition to the publication 'The Snark, a counterexample for Church's thesis?' examples and details are offered in the form of two appendices C6 and C7 that allow for better understanding of the general method and the particular problem related to Church's thesis. The author has developed an approach to logics that comprises, but also 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 etc. The most usual approach to calculative (effectively calculable) functions is done by register machines or similar storage-based computers like the Abacus or Turing machines. Another usual approach to computable functions is to start with primitive recursive functions. However, one has to find a way to put this into a form that does not rely on a pre-knowledge about functions and higher logic. The concrete calcule LAMBDA of decimal pinitive arithmetic allows for such an access. It is based on a machine that is completely different from the storage-based machines: the PINITOR does not use storages but rather many microprocessors, one for each appearance of a command in the code of primitive recursive functior. the codes are decimal numbers, called pinons , where only the characters 0 1 2 8 9 appear. There are four kind of commands only: 0 nullification, 1 succession, 2 straight recursion and 8 composition. The PINITOR is a calculator which means that there is no halting problem. Computers have halting problems, per defintion calculators do not. Appendix C6 gives the programming of the codes of most of the usual primitive function and goes even farther, e.g. it introduces generator technique that allows for the straight-forward calculation of so-called processive function, that are not primitive recursive. The most famous examples of Ackermann and other hyperexponential functions are programmed. Appendix C7 turns to the Boojum-function and the Snark-function that have been introduced as calculative functions in the above publication in connection with Church's thesis. A list is provided that gives the lowest values for these functions and gives some more insight into these functions.

Comments: 17 Pages. The details given here relate to document viXra: 1905.0221

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[v1] 2019-05-17 10:56:36

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