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

*<sup>I</sup>*. Let the form *<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 *<sup>d</sup>*R ¼ <sup>P</sup>*<sup>n</sup> i*¼1 1 *<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>*Xi*d*xi* <sup>¼</sup> 0 has the solution in 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** *P x*<sup>0</sup> <sup>1</sup> , … , *x*<sup>0</sup> *n* � � **determined by our choice of** const. = *C***. Only the points lying in the hyperplane** <sup>R</sup> *<sup>x</sup>*<sup>0</sup> <sup>1</sup> , … , *x*<sup>0</sup> *n* � � **are accessible from the point** *P* **along the path satisfying 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**).

*II*. Let us pick the point *V*, e.g., from <sup>3</sup> , lying in a vicinity of the point *P*, which is not accessible from *P* following the path d*Q* = 0. Let *g* be a line going through the point *P* and let *g* be oriented ( *g* !) in such way that it does not satisfy the condition d*Q* = 0. The point *V* and the line *g* determine a plane *Xi* = *Xi*(*u, v*), *i* = 1*,* 2*,* 3. Let us

**Figure A2.** *The proof way of the Caratheodory theorems.*

consider a curve *k* in this plane, going through the point *V* (*u*0*, v*0) in that way ( *g* !) that d*Q* = 0 is supposedly valid along this curve. **There is only one curve** *k* **for the point** *V* (*u*0*, v*0). It lies in our plane, the plane *Xi* = *Xi*(*u, v*), and then it is valid for it d*Xi* <sup>¼</sup> *<sup>∂</sup>Xi <sup>∂</sup><sup>u</sup>* <sup>d</sup>*<sup>u</sup>* <sup>þ</sup> *<sup>∂</sup>Xi <sup>∂</sup><sup>v</sup>* <sup>d</sup>*<sup>v</sup>* and, considering d*<sup>Q</sup>* = 0 along *<sup>k</sup>*, we get <sup>P</sup><sup>3</sup> *<sup>i</sup>*¼<sup>1</sup>*Xi ∂Xi <sup>∂</sup><sup>u</sup>* d*u*þ P<sup>3</sup> *<sup>i</sup>*¼<sup>1</sup>*Xi ∂Xi <sup>∂</sup><sup>v</sup>* d*v* ¼ 0.

*ΔA*<sup>00</sup> ¼ *ΔQ*<sup>00</sup>

*ΔA*00 k*T*<sup>00</sup> *W*

> *ΔA* k*TW*

*<sup>Δ</sup>A*<sup>∗</sup> <sup>¼</sup> *<sup>Δ</sup><sup>A</sup>* � *<sup>Δ</sup>A*<sup>00</sup>

of the whole information entropy *H*\*

k*TW*

its working temperatures *TW > T <sup>W</sup>* = *T*\*

*<sup>β</sup>*<sup>00</sup> <sup>¼</sup> *<sup>Δ</sup>Q*<sup>00</sup> 0 *ΔQ*<sup>00</sup> *W* ¼

*<sup>β</sup>* <sup>¼</sup> *<sup>Δ</sup>Q*<sup>0</sup> *ΔQW* ¼

> *W TW*

<sup>¼</sup> *<sup>T</sup>*<sup>∗</sup> 0 *TW*

≜*β* <sup>∗</sup>

*β <sup>β</sup>*<sup>00</sup> <sup>¼</sup> *<sup>T</sup>*<sup>00</sup>

**19**

<sup>¼</sup> *H X*ð Þ� <sup>1</sup> � *<sup>T</sup>*<sup>0</sup>

temperature *TW*. Evidently, the sense of the symbol *T*<sup>00</sup>

*ΔQ*<sup>00</sup> 0 *T*<sup>00</sup> *W ΔQ*<sup>00</sup> *W T*<sup>00</sup> *W*

*ΔQ*<sup>0</sup> *TW ΔQW TW*

OO<sup>00</sup> and when <sup>Δ</sup>*Q*<sup>0</sup> <sup>=</sup> <sup>Δ</sup>*<sup>Q</sup>* 0) is expressible by the symbol *<sup>T</sup>*\*

*<sup>H</sup>*<sup>∗</sup> *<sup>Y</sup>* <sup>∗</sup> ð Þ¼ *<sup>Δ</sup>A*<sup>∗</sup>

*<sup>W</sup>* � <sup>1</sup> � *<sup>T</sup>*<sup>0</sup> *T*00 *W* 

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

*W TW* � *<sup>T</sup>*<sup>0</sup> *TW* 

> *<sup>W</sup>* <sup>1</sup> � *<sup>T</sup>*<sup>0</sup> *T*0 *W*

and thus, for the cycles O00and O, it is valid that

*T*<sup>00</sup> *W*

*TW* 

¼ k*TW* � *H X*ð Þ� ð Þ� 1 � *β kT*<sup>00</sup>

(*Y*\*

<sup>¼</sup> *H X*ð Þ� ð Þ� <sup>1</sup> � *<sup>β</sup> <sup>T</sup>*<sup>00</sup>

� *<sup>T</sup>*<sup>00</sup> *W TW* þ *T*<sup>0</sup> *TW*

*TW*

the working temperatures of the whole cycle OO<sup>00</sup> are *TW* and *<sup>T</sup> <sup>W</sup>* <sup>=</sup> *<sup>T</sup>*\*

<sup>¼</sup> *H Y*00j*X*<sup>00</sup> ð Þ

<sup>¼</sup> *H X*ð Þ <sup>j</sup>*<sup>Y</sup>*

Then, for the whole change of the thermodynamic entropy within the combined cycle OO<sup>00</sup> (measured in information units *Hartley, nat, bit*) and thus for the change

It is valid, for Δ*A*\* is a *residuum work* after the work Δ*A* has been performed at the

(30) expresses that fact that the double cycle OO00 is the direct Carnot Cycle just with

*H Y*<sup>00</sup> ð Þ <sup>¼</sup> *<sup>T</sup>*<sup>0</sup>

*H X*ð Þ <sup>¼</sup> *<sup>T</sup>*<sup>0</sup>

*T*<sup>00</sup> *W* , *T*<sup>00</sup>

*TW*

), it is valid that

*W TW* � <sup>1</sup> � *<sup>β</sup>*<sup>0</sup> ð Þ

For the whole work <sup>Δ</sup>*A*\* of the combined cycle OO00

<sup>¼</sup> *H X*ð Þ� <sup>1</sup> � *<sup>T</sup>*<sup>0</sup>

<sup>¼</sup> *H X*ð Þ� <sup>1</sup> � *<sup>T</sup>*<sup>0</sup>

and, further, for Δ*A* in the cycle O, we have

<sup>¼</sup> *<sup>Δ</sup>QW* � *<sup>T</sup>*<sup>00</sup>

¼ k � *H X*ð Þ� *T*<sup>00</sup>

¼ *ΔQW:*

*Common Gnoseological Meaning of Gödel and Caratheodory Theorems*

*T*00 *W TW*

¼ k � *H X*ð Þ� *T*<sup>00</sup>

¼ k � *H X*ð Þ� *T*<sup>00</sup>

*<sup>Δ</sup><sup>A</sup>* <sup>¼</sup> <sup>k</sup> � *H X*ð Þ� *TW*ð Þ¼ <sup>1</sup> � *<sup>β</sup>* <sup>k</sup> � *H X*ð Þ� *TW* <sup>1</sup> � *<sup>T</sup>*<sup>0</sup>

� <sup>1</sup> � *<sup>T</sup>*<sup>0</sup> *T*00 *W* 

> *<sup>W</sup>* � *T*<sup>0</sup>

¼

<sup>¼</sup> *H X*ð Þ� <sup>1</sup> � *<sup>β</sup>*<sup>00</sup> <sup>¼</sup> *H X*ð Þ� *<sup>η</sup>*<sup>00</sup>

¼ *H X*ð Þ� ð Þ¼ 1 � *β H X*ð Þ� *ηmax*

*<sup>W</sup>* � *H X*ð Þ� <sup>1</sup> � *<sup>β</sup>*<sup>00</sup> <sup>&</sup>gt;0 (29)

0. In the double cycle OO<sup>00</sup>

*<sup>W</sup>* <sup>¼</sup> *<sup>T</sup>*<sup>∗</sup>

, cyklus O

*<sup>W</sup>* <sup>1</sup> � *<sup>β</sup>*<sup>00</sup> <sup>¼</sup> <sup>k</sup> � *<sup>T</sup>*<sup>00</sup>

*<sup>W</sup>* � *H Y*<sup>00</sup>

*TW*

*max*

, we have

<sup>¼</sup> *H X*ð Þ� <sup>1</sup> � *<sup>T</sup>*<sup>00</sup>

*W TW* 

*<sup>W</sup>* (within the double cycle

0, cyklus O00

0, which is possible, for

, it is valid that

0. The relation

(31)

(27)

(26)

(28)

(30)

**The curve** *k***, however, intersects the line** *g* **in the point** *R*, **which is inaccessible from the point** *P* **along the path with** d*Q* = 0 (for d*QR <sup>g</sup>* ! ¼6 0Þ. **Otherwise, the**

**point** *<sup>V</sup>* **would also be accessible from the point** *<sup>P</sup>* **through** *<sup>R</sup>* **and** *<sup>k</sup>* <sup>d</sup>*QRk* <sup>¼</sup> <sup>0</sup> � �, **which is a conflict with the original assumption.** By a suitable selection of *V*, it is possible to have the point *R* arbitrarily close to the point *P*; in the arbitrary vicinity of the point *P,* there are points inaccessible from the point *P* along the path with d*Q* = 0. Now, let us pick a line *g*<sup>0</sup> parallel to the line *g*, and a cylinder *C* going through these two lines. We consider that the curve *k* satisfying the relation d*Q* = 0 is on this cylinder *C*<sup>0</sup> goes through the point *P* and intersects the line *g*<sup>0</sup> in the point *M*.

Now, let us consider another cylinder *C*<sup>0</sup> as the continuation of *C* with *g*<sup>0</sup> and *g*. Let us use the symbol *k*<sup>0</sup> for the continuation of the curve *k* in *C*<sup>0</sup> . Then the curve *k*<sup>0</sup> must intersect the line *g* in the point *P*. Otherwise, it would be possible to deform the plane *C*<sup>0</sup> as much as to get *C*, thus continually merging the intersecting point *N* into the point *P* and at the moments of discrepancy of the points *P* and *N*, it would be possible to reach the point *P* from the point *N* along the line *g* (supposedly with d*Q* = 0). However, the condition d*Q* = 0 is not valid there (d*QR <sup>g</sup>* ! 6¼ 0Þ. By deforming *C*<sup>0</sup> into *C*, the *k* and *k*<sup>0</sup> would close a plane *F* where d*Q* = 0. If the equation of this plane has the form Rð Þ *xi* 3 *<sup>i</sup>*¼<sup>1</sup> <sup>¼</sup> const*:*, then the equation d*<sup>Q</sup>* = 0 has a solution—an integration factor for the **Pfaff form** *<sup>δ</sup><sup>Q</sup>* <sup>¼</sup> <sup>P</sup><sup>3</sup> *<sup>i</sup>*¼<sup>1</sup>*Xi*d*xi* exists [11].

## **A.3 Information thermodynamic concept removing autoreference**

The concept for ceasing the autoreference, based on the two Carnot Cycles disconnected as for their heaters and described informationally, shows the following **Figure A3**. (also see [1, 2, 4]):

For Δ*A ,* it is valid in the cycle O00 that

**Figure A3.** *The concept for ceasing the autoreference.*

*Common Gnoseological Meaning of Gödel and Caratheodory Theorems DOI: http://dx.doi.org/10.5772/intechopen.87975*

$$
\Delta A'' = \Delta Q\_W'' \cdot \left( \mathbf{1} - \frac{T\_0}{T\_W''} \right) = \Delta Q\_W \cdot \frac{T\_W''}{T\_W} \cdot \left( \mathbf{1} - \frac{T\_0}{T\_W''} \right) = 
$$

$$
= \Delta Q\_W \cdot \left( \frac{T\_W''}{T\_W} - \frac{T\_0}{T\_W} \right) = \mathbf{k} \cdot H(\mathbf{X}) \cdot \left( T\_W'' - T\_0 \right) \tag{26}
$$

$$
= \mathbf{k} \cdot H(\mathbf{X}) \cdot T\_W'' \left( \mathbf{1} - \frac{T\_0}{T\_W'} \right) = \mathbf{k} \cdot H(\mathbf{X}) \cdot T\_W'' \left( \mathbf{1} - \boldsymbol{\beta}^{\*'} \right) = \mathbf{k} \cdot T\_W' \cdot H(\mathbf{Y}^{\*'})
$$

and, further, for Δ*A* in the cycle O, we have

$$
\Delta A = \mathbf{k} \cdot H(\mathbf{X}) \cdot T\_W (\mathbf{1} - \beta) = \mathbf{k} \cdot H(\mathbf{X}) \cdot T\_W \left( \mathbf{1} - \frac{T\_0}{T\_W} \right) \tag{27}
$$

and thus, for the cycles O00and O, it is valid that

$$\begin{aligned} \frac{\Delta A^{\prime\prime}}{\mathbf{kT}\_{W}^{\prime\prime}} &= H(\mathbf{X}) \cdot \left(\mathbf{1} - \frac{T\_0}{T\_W^{\prime\prime}}\right) = H(\mathbf{X}) \cdot \left(\mathbf{1} - \boldsymbol{\beta}^{\prime\prime}\right) = H(\mathbf{X}) \cdot \eta\_{\max}^{\prime\prime} \\ \frac{\Delta A}{\mathbf{kT}\_W} &= H(\mathbf{X}) \cdot \left(\mathbf{1} - \frac{T\_0}{T\_W}\right) = H(\mathbf{X}) \cdot \left(\mathbf{1} - \boldsymbol{\beta}\right) = H(\mathbf{X}) \cdot \eta\_{\max} \end{aligned} \tag{28}$$

For the whole work <sup>Δ</sup>*A*\* of the combined cycle OO00 , we have

$$
\Delta A^\* = \Delta A - \Delta A^{\prime} = \left[ \mathbf{k} T\_W \cdot H(\mathbf{X}) \cdot (\mathbf{1} - \boldsymbol{\beta}) - \mathbf{k} T\_W^{\prime} \cdot H(\mathbf{X}) \cdot (\mathbf{1} - \boldsymbol{\beta}^{\prime}) \right] > \mathbf{0} \tag{29}
$$

Then, for the whole change of the thermodynamic entropy within the combined cycle OO<sup>00</sup> (measured in information units *Hartley, nat, bit*) and thus for the change of the whole information entropy *H*\* (*Y*\* ), it is valid that

$$\begin{split} H^\*(Y^\*) &= \frac{\Delta \mathcal{A}^\*}{\mathbf{k} T\_W} = H(\mathbf{X}) \cdot \left[ (\mathbf{1} - \boldsymbol{\beta}) - \frac{T\_W^\prime}{T\_W} \cdot (\mathbf{1} - \boldsymbol{\beta}^\prime) \right] \\ &= H(\mathbf{X}) \cdot \left( \mathbf{1} - \frac{T\_0}{T\_W} - \frac{T\_W^\prime}{T\_W} + \frac{T\_0}{T\_W} \right) = H(\mathbf{X}) \cdot \left( \mathbf{1} - \frac{T\_W^\prime}{T\_W} \right) \end{split} \tag{30}$$

It is valid, for Δ*A*\* is a *residuum work* after the work Δ*A* has been performed at the temperature *TW*. Evidently, the sense of the symbol *T*<sup>00</sup> *<sup>W</sup>* (within the double cycle OO<sup>00</sup> and when <sup>Δ</sup>*Q*<sup>0</sup> <sup>=</sup> <sup>Δ</sup>*<sup>Q</sup>* 0) is expressible by the symbol *<sup>T</sup>*\* 0, which is possible, for the working temperatures of the whole cycle OO<sup>00</sup> are *TW* and *<sup>T</sup> <sup>W</sup>* <sup>=</sup> *<sup>T</sup>*\* 0. The relation (30) expresses that fact that the double cycle OO00 is the direct Carnot Cycle just with its working temperatures *TW > T <sup>W</sup>* = *T*\* 0. In the double cycle OO<sup>00</sup> , it is valid that

$$\begin{aligned} \beta'' &= \frac{\Delta Q\_0''}{\Delta Q\_W''} = \frac{\frac{\Delta Q\_0''}{T\_W''}}{\Delta Q\_W''} = \frac{H(Y''|X'')}{H(Y'')} = \frac{T\_0}{T\_W''}, T\_W'' = T^\* \,\_0, \text{ cyklus } \mathcal{O}''\\ \beta &= \frac{\Delta Q\_0}{\Delta Q\_W} = \frac{\frac{\Delta Q\_0}{T\_W}}{\frac{\Delta Q\_W}{T\_W}} = \frac{H(X|Y)}{H(X)} = \frac{T\_0}{T\_W}, \text{cyklus } \mathcal{O} \end{aligned} \tag{31}$$
 
$$\begin{aligned} \frac{\beta}{\beta''} &= \frac{T'}{T\_W} = \frac{T\_0^\*}{T\_W} \triangleq \beta^\* \end{aligned} \tag{32}$$

and then, by (30) and (31) is writable that

$$\frac{\Delta A^\*}{\text{kT}\_W} = H(\mathbf{X}) \cdot (\mathbf{1} - \boldsymbol{\beta}^\*) = H(\mathbf{X}) \cdot \left[ \mathbf{1} - \frac{H(\mathbf{X}|\mathbf{Y}) \cdot H(\mathbf{Y}'')}{H(\mathbf{Y}''|\mathbf{X}'') \cdot H(\mathbf{X})} \right] > 0\tag{32}$$

It is ensured by the propositions *TW > T <sup>W</sup>*,*T* <sup>0</sup> = *T*<sup>0</sup> and also by that fact that the loss entropy *H*(*X*|*Y*) is described and given by the heat Δ*Q*<sup>0</sup> = Δ*Q*″ 0. But in our combined cycle OO00 , it is valid too that

$$H(X) = \frac{\Delta Q\_W}{\text{kT}\_W} = \frac{\Delta Q\_W}{\text{kT}\_W^{\prime}} = H(Y^{\prime}) \quad \left[= \frac{\Delta Q\_W^{\prime\prime}}{\text{kT}^\* \, ^\circ \text{o}}\right] \tag{33}$$

and we have

$$\frac{H(\mathbf{X}|\mathbf{Y})}{H(\mathbf{Y}''|\mathbf{X}'')} = \boldsymbol{\beta}^\* < \mathbf{1} \tag{34}$$

For the whole information entropy *<sup>Δ</sup>A*<sup>∗</sup> <sup>k</sup>*TW* (the whole thermodynamic entropy SC in information units) and by following the previous relations also it is valid that

$$\begin{split} \frac{\Delta A^\*}{\text{k}T\_W} &= H(Y'') - H(Y'') \cdot \boldsymbol{\beta}^\* = H(Y'') \cdot \left( \mathbf{1} - \frac{T\_0^\*}{T\_W} \right) \\ &= H(Y'') \cdot \left[ \mathbf{1} - \frac{H(X|Y)}{H(X''|Y'')} \right] \end{split} \tag{35}$$

And thus, the structure of the information transfer channel K [expressed by the quantity *H*(*X*|*Y*)] is measurable by the value *H*\* (*Y*\* ) from (32) and (35). Symbolically, we can write, using a certain growing function *f*,

$$H^\*\left(Y^\*\right) = \frac{\Delta A^\*}{\mathbf{k}T\_W} \cong f[H(X|Y)] > \mathbf{0} \tag{36}$$

**Author details**

Department of Mathematics, University of Chemistry and Technology, Prague,

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: hejnab@vscht.cz

*Common Gnoseological Meaning of Gödel and Caratheodory Theorems*

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

provided the original work is properly cited.

Bohdan Hejna

Czech Republic

**21**

The cycles O, O<sup>00</sup>, and OO00 are the Carnot Cycles, and thus from their definition and construction, they are imaginatively<sup>16</sup> in principle, the infinite cycles; in each of them the following *criterion of an infinite cycle* (see [12]) it is valid inevitably,

$$H\left(X^{[\cdot]};Y^{[\cdot]}\right) = H\left(X^{[\cdot]}\right) - H\left(X^{[\cdot]}|Y^{[\cdot]}\right) = H\left(Y^{[\cdot]}\right) > 0 \text{ and } \ \Delta S\_{\mathcal{L}}^{[\cdot]} = 0 \tag{37}$$

The construction of the cycle OO<sup>00</sup> enables us to recognize that the infinite cycle O is running. In our case, it is the infinite cycle from (5), (6) and also from [4, 8, 10],

$$\begin{array}{ccccccccc} Q(\mathbb{X}, \ Y), & Q[\mathbb{X}, \ \Phi[Q(\mathbb{X}, \ Y)]], & Q[\mathbb{X}, \ \Phi[Q(\mathbb{X}, \ \Phi[Q(\mathbb{X}, \ Y)])]], & \dots \\ Q(\mathbb{N}\_{0}, \ Y), & Q[\mathbb{N}\_{0}, \Phi[Q(\mathbb{N}\_{0}, \ Y)]], & Q[\mathbb{N}\_{0}, \Phi[Q(\mathbb{N}\_{0}, \ \Phi[Q(\mathbb{N}\_{0}, \ Y)])]], & \dots \end{array} \tag{38}$$

<sup>16</sup> When an infinite reserve of energy would exist.

*Common Gnoseological Meaning of Gödel and Caratheodory Theorems DOI: http://dx.doi.org/10.5772/intechopen.87975*
