Analysis shows the formals and mathematical basis on which the incompleteness theorem reposes qua limitation theorem which is a part of fundamental science. The duality of theoretical and metatheoretical levels in fundamental science allows to establish the mathematical been. In this point, the mathematical mind is penetrable if the critical cogitation lean on formals systems.

The degree of freedom of a machine is organized and specified by a human been. Even it’s well known that a machine is very quick and very reliable than a human been; this remain in the center because the algorithm was made by him. The calculable functions in TURING’s machine are all calculable in intuitive point of view. So there are the undecidable properties in all axiomatic for formalizing the arithmetic.

The internal limitations of relativist quantic mechanic establish the incomplete character and undecidable of physics theories in the light of results of GÖDEL’s theorem. From this mechanic, the research shows that a formal system which involves the natural integers cannot be in the same time complete and consistent. In addition, it proves that the coherence of a such system can be demonstrate “inside” of formal system.

The GÖDEL’s theorem is intrinsically a theorem of limitation of the formals systems. The theorem shows that the coherence of PEANO’s arithmetic cannot be demonstrate by a simple way. This constitutes an opposite shock in metamathematic design in HILBERT’s perspective. Finally, if we want a proof of arithmetic coherence, it is sufficient to approve the arbitrary notions the type of function and function of function, and that next to concretes symbols.

The research shows clearly that the HILBERT’s program look at to obtain the formalisms of the formals theories. This program stamp a fertile optimism of symbolic which allowed to create the logics and mathematics formalisms, and there formal and automatable manipulation. So the formalism is the image of thinking; in this sense the forms become the work matter.

By applying the formal arithmetic model, the mathematic and logic theories have created the self-references and have chosen a specific model for accomplish the logic-mathematics proves. A formal arithmetic (ROBINSON et PEANO) constitute in this fact the basic hypothesis for the two incompleteness theorems.

The resort in arithmetization of arithmetic reaches to construct inside of arithmetic a proposition which confirm its self-indemonstrability. In substance, the proof of coherence presupposes a form of induction in transfinite order for proving the coherence of arithmetic which is the finite order. It is dealing recursive functions which have the properties for all attribute values system to determine them by the means of finite type procedure. It comes to recursive arithmetic, to translate the elements of metatheory formal system.

The study presents systematically the formals and theoretical foundations of incompleteness theorem: framework, hypothesis, rules of provability. The recursive functions constitute the operational foundations in the development, the construction and the prove of this theorem. The preparatory theorems and the sense theorem are the socle which establishes the incompleteness.

The provability of a formal system is an insufficient criterion to translate properly the truth notion of logico-mathematic. The incompleteness is well understanding by its reference in interpretation and truth concept is one of the result. Taking into account the level of language -object language and metalanguage- allow skirting interns contradictions and establishing the logic consistence of the formal system.

Of the point of view mathematical-logic, the researches specify the intrinsic limits connected on knowledge apprehensions. In regard of POPPER’s refutability, the incompleteness misses a significant come-back in the way of knowledge apprehend. In fact, it releases a significant insufficiency of induction and that of verification. So it will subvert an universal character of certainty.The rationational thruth and the true rationality are linked in a metasystem .They are depending on a system which will be in the same time empiricaly proved and logicaly assured.