**Authors:** Hannes Hutzelmeyer

In 1936 Alonzo Church put forward his thesis that recursive functions comprise all effectively calculative functions. Whereas recursive functions are precisely defined, effectively calculative functions cannot be defined with a rigor that is requested by mathematicians. There has been a considerable amount of talking about the plausibility of Church's thesis, however, this is not relevant for a strict mathematical analysis. The only way to end the discussion is obtained by a counterexample. The author has developed an approach to logics that comprises, but 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 and so on. The concrete calcule LAMBDA of natural number arithmetic with first-order logic has been defined by the author. It includes straight recursion and composition of functions, it contains a wide range of so-called compinitive functions, with processive functions far beyond primitive recursive functions. All recursive functions can be represented in LAMBDA too. The unary Snark-function is defined by a diagonalization procedure such that it can be calculated in a finite number of steps. However, this calculative function transcends the compinitive functions and presumably the recursive functions. The defenders of Church's thesis are challenged to show that the Snark-function is recursive. Another challenge asks for an example of a recursive function that cannot be expressed as a compinitive function, i.e. without minimization.

**Comments:** 7 Pages.

**Download:** **PDF**

[v1] 2019-05-16 03:15:48

**Unique-IP document downloads:** 13 times

Vixra.org is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary. In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution. Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.

**Add your own feedback and questions here:**

*You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful. *