**5. Conclusion**

Peano Arithmetic theory is generated by its inferential rules (rules of the inferential system in which it is formulated). It consists of parts bound mutually just by these rules, but none of them is not identical with it nor with the system in their totality.

By information-thermodynamic and computing analysis of Peano arithmetic proving, we have showed why the Gödel formula and its negation are not provable and decidable within it. They are constructed, not inferred, by the diagonal argument, which is not from the set of the inferential rules of the system. The attempt to prove them leads to awaiting of the end of the infinite cycle being generated by the application of the substitution function just by the diagonal argument. For this case, the substitution function is not countable and for this it is not recursive (although in the Gödel original definition is claimed that it is). We redefine it to be total by the zero value for this case. This new substitution function generates the Gödel numbers of chains, which are not only satisfying the recursive grammar of formulae but it itself is recursive. The option of the zero value follows also from the vision of the inferential process as it would be the information transfer. The attempt to prove the Gödel Undecidable Formula is the attempt of the transfer of that information, which is equal to the information expressing the inner structure of the information transfer channel. In the thermodynamic point of view, we achieve the equilibrium status, which is an equivalent to the inconsistent theory. So, we can see that the Gödel Undecidable Formula is not a formula of the Peano Arithmetics and, also, that it is not an arithmetical claim at all. From the thermodynamic consideration follows that even we need a certain effort or energy to construct it, within the frame of the theory this is irrelevant. It is the error in the inference and cannot be part of the theory and also it is not the system. Its information value in it (as in the system of the information transfer) is zero. But it is the true claim about **inferential properties of the theory** (in fact, of the **properties of the information transfer**).

Any description of real objects, no matter how precise, is only a model of them, of their properties and relations, making them available in a specified and somewhat limited (compared with the reality) point of view determined by the description/ model designer. This determination is expressed in definitions and axiomatics of this description/model/theory—both with definitions and by axioms and their number. Hence, realistically/empirically or rationally, it will also be true about (objects of) reality what such a model, called *recursive and able-of-axiomatization*, does not include. With regard of reality any such a model is *axiomatically incomplete*, even if the system of axioms *is complete*. **In addition, and more importantly, this description/model of objects, of their properties and possible relations** (the theory about reality) **cannot include a description of itself** just as the object of reality defined by itself (any such theory/object is not a subject of a direct description of itself). The description/model or the theory about reality is a grammar construction with substitutes and axiomatization and, as such, it is *incomplete in the Gödelian way*—**the grammar itself does not prevent a semantical mixing; but any observed real object cannot be the subject of observation of itself and this is valid for the considered theory, just as for the object of reality, too**. No description of reality arranged from its inside or created within the theory of this reality can capture the reality completely in wholeness of its all own properties. It is impossible for the models/theories considered, independently on their axiomatization. They are limited in principle [in the *real sense* of the Gödel theorems (in the Gödelian way)].

Now, **with our better comprehension**, we can claim that the **consistency of the recursive and axiomatizable system can never be proved in it itself**,

even if the system is consistent really. The reason is that a **claim of the consistency of such a system is designable only if the system is the object of** *outer observation/measuring/studies***, which is not possible within the system itself**. Ignoring this approach is also the reason for the formulation of the **Gibbs paradox** and **Halting Problem**. Also, our awareness of this fact results in our **full understanding** of the **meaning and proof of the Gödel theorems**, very often explained and described incomprehensibly, even inconsistently or paradoxically, **and which is parallel with the way of the Caratheodory proof of the** *II.* **Thermodynamic Principle**.

♣♦◊

**A.2 The proof way of Caratheodory theorems**

*Common Gnoseological Meaning of Gödel and Caratheodory Theorems*

*DOI: http://dx.doi.org/10.5772/intechopen.87975*

*<sup>v</sup>Xi*d*xi*. Then the Pfaff equation *<sup>δ</sup><sup>Q</sup>* <sup>¼</sup> <sup>P</sup>*<sup>n</sup>*

*<sup>i</sup>*¼<sup>1</sup>*Xixi* has the integration factor *<sup>v</sup>* and let

� � **are accessible from the point** *P* **along the path satisfy-**

the form Rð Þ¼ *x*1, … , *xk* const. and this solution represents a family of hyperplanes in *n*-dimensional space, not intersecting each other. **Let us pick now the point**

� � **determined by our choice of** const. = *C***. Only the points lying in the**

**ing the condition** d*Q* = 0**. All the points not lying in this hyperplane are inaccessible from the point** *P* **along the path satisfying the condition** d*Q* =0(**Figure A2**).

is not accessible from *P* following the path d*Q* = 0. Let *g* be a line going through the

d*Q* = 0. The point *V* and the line *g* determine a plane *Xi* = *Xi*(*u, v*), *i* = 1*,* 2*,* 3. Let us

*<sup>i</sup>*¼<sup>1</sup>*Xi*d*xi* <sup>¼</sup> 0 has the solution in

, lying in a vicinity of the point *P*, which

!) in such way that it does not satisfy the condition

*<sup>I</sup>*. Let the form *<sup>δ</sup><sup>Q</sup>* <sup>¼</sup> <sup>P</sup>*<sup>n</sup>*

*n*

point *P* and let *g* be oriented ( *g*

*The proof way of the Caratheodory theorems.*

<sup>1</sup> , … , *x*<sup>0</sup>

*Example of not distinguishing the reality and its image.*

*II*. Let us pick the point *V*, e.g., from <sup>3</sup>

*n*

*<sup>d</sup>*R ¼ <sup>P</sup>*<sup>n</sup>*

**Figure A1.**

**Figure A2.**

**17**

*P x*<sup>0</sup>

*i*¼1 1

<sup>1</sup> , … , *x*<sup>0</sup>

**hyperplane** <sup>R</sup> *<sup>x</sup>*<sup>0</sup>
