**2. Gödel theorems**

$$\begin{aligned} \textbf{Remark: The expression } & \mathrm{Sb} \begin{pmatrix} u\_1 & u\_2 \\ t & \\ Z(\mathbf{x}) & Z(\mathbf{y}) \end{pmatrix} \text{ or the expression} \\ \textbf{Sub} \begin{pmatrix} 17 & 19 \\ t & \\ Z(\mathbf{x}) & Z(\mathbf{y}) \end{pmatrix} \text{ represents the result value of the Gödel number } t[Z(\mathbf{x}), Z(\mathbf{y})], \text{ and } \mathbf{x} \text{ is the } \mathbf{x} \text{-direction.} \\ \textbf{Sub} \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix} \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix} \end{aligned}$$

which is coding the (constant) claim *T* (*x, y*) z *PM* has been generated by the substitution of *x* a *y* instead of the free variables *X* and *Y* in the function *T* (*X, Y*) from *PM* with its Gödelian code *t*(*u*1*, u*2) in the (arithmetized) P,

$$\operatorname{Sb}\begin{pmatrix} & u\_1 & u\_2 \\ & & \\ & Z(\mathbf{x}) & Z(\mathbf{y}) \end{pmatrix} = \operatorname{Sb}\begin{pmatrix} & \mathbf{1}\mathcal{T} & & \mathbf{1}\mathcal{Y} \\ & & \\ & Z(\mathbf{x}) & Z(\mathbf{y}) \end{pmatrix}.$$

♣ **Into the** *VARIABLES,* **we substitute the** *SIGNS* **of the same** *type* **but the introduction of the term** *admissible substitution* **itself is not supposing it wordly.**

**- Then it is possible to work even with the expressions not grammatically correct and thus with such chains, which are not** *FORMULAE* **of the system** P **(and thus not belonging into the theory** T PA**).**

**Then the substitution function** *Sb* <sup>⋯</sup> � ⋯ ! **is not possible, within the frame**

**of the inference in the system** P**, be used isolately as an arbitrarily performed number manipulation—**in spite of the fact that it is such number manipulation really. **It is used only and just within the frame of the language** LP **and, above all, within the frame of the conditions specified by the právě a jenom**

<sup>5</sup> *Form*ula, *R*eihe von *F*ormeln, *Op*eration, *F*o*l*ge, *Gl*ied, *B*eweis, *Bew*eis, see Definition 1–46 in [5–7] and by means of all other, by them 'called', relations and functions (by their procedures).

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

*INFERENCE* **of the elements of the language** LT PA only (and thus in the more limited way).

Others than/semantically (or by the type) homogenous application of the substitution function is not within the right inference/INFERENCE within the system <sup>P</sup> possible.<sup>6</sup>

#### **2.1 The Gödel** *UNDECIDABLE CLAIM***'s construction**

♦ Let the Gödel numbers *<sup>x</sup>* and *<sup>y</sup>* be given. The number *<sup>x</sup>* is the *SEQUENCE OF FORMULAE* valid and *y* is a *FORMULA* of P. We define the valid constant relation *Q*(*x, y*) from the *Q*(*X, Y*) for given values *x* and *y*, *X*:=*x*, *Y*:= *y*; 17 = Φ(*X*), 19 = Φ(*Y*),7,8

$$Q(\mathbf{x}, y) \equiv \overline{\mathbf{x} \mathbf{B}\_{\mathbf{x}} \left[ \mathbf{S} \mathbf{b} \begin{pmatrix} 19 \\ y \\ Z(p) \end{pmatrix} \right]} \equiv \text{Bew}\_{\mathbf{x}} \left[ \mathbf{S} \mathbf{b} \begin{pmatrix} 17 & 19 \\ q \\ Z(\mathbf{x}) & Z(\mathbf{y}) \end{pmatrix} \right] \tag{1}$$
 
$$q[Z(\mathbf{x}), \ Z(\mathbf{y})] = \Phi[Q(\mathbf{x}, \ \mathbf{y})], \ \overline{\mathbf{x} \mathbf{B}\_{\mathbf{x}} \mathbf{y}} \equiv \text{Bew}\_{\mathbf{x}}(\mathbf{y}') = \text{Bew}\_{\mathbf{x}}[y[Z(\mathbf{y})]] = \text{Bew}\_{\mathbf{x}}[q[Z(\mathbf{x}), \ \mathbf{Z}(\mathbf{y})]] \tag{1}$$

♦ Now we put *<sup>p</sup>* = 17*Gen q*, *<sup>q</sup>* <sup>=</sup> *<sup>q</sup>*(17, 19) *<sup>q</sup>*ð Þ 17, 19 <sup>≜</sup> *Q X*ð Þ , *<sup>Y</sup>* � � and then,

$$p\_{\perp} = \text{17Gen}q(\mathbf{17}, \mathbf{19}) = \Phi[\mathbb{V}\_{\mathbf{x} \in \mathbb{X}} | Q(\mathbf{x}, \mathbf{Y})] \triangleq Q(\mathbf{X}, \mathbf{Y}) \triangleq Q(\mathbb{N}\_0, \mathbf{Y})\tag{2}$$

The meta-language symbol *Q*ð Þ , *Y* or *Q*ð Þ 0, *Y* is to be read: **No** *x*∈ ð Þ<sup>0</sup> **is in the** *κ*-*INFERENCE* **relation to the variable** *Y* (to its space of values ).

♦ Further, with the Gödel substitution function, we put *<sup>q</sup>*[17*, Z*(*p*)] = *<sup>r</sup>*(17) = *<sup>r</sup>*,

$$r \coloneqq \text{Sb}\begin{pmatrix} 19 \\ q \\ Z(p) \end{pmatrix} \text{ and then} \\ r = \text{Sb}\begin{pmatrix} 19 \\ q(17, 19) \\ Z(p) \end{pmatrix} = r(17) = \Phi[\text{Q}(X, \ p)] \tag{3}$$

The Gödel number *r* is, by the substitution of the *NUMERAL Z*(*p*), **supposedly only** (by [5–7]) the *CLASS SIGN* with the *FREE VARIABLE* 17 (*X*); with the values *p,* the *r* contains the feature of *autoreference*,

$$\begin{aligned} r &= r(\mathbf{1}\mathbf{7}) = q[\mathbf{1}\mathbf{7}, \ \operatorname{Z}[p(\mathbf{1}\mathbf{9})]] = q[\mathbf{1}\mathbf{7}, \ \operatorname{Z}[\mathbf{1}\mathbf{7}\operatorname{Gen}q(\mathbf{1}\mathbf{7}, \mathbf{19})]] \triangleq \operatorname{Q}(\mathbf{X}, \ p) \\\ &= \Phi[\operatorname{Q}[\mathbf{X}, \ \operatorname{\Phi}[\forall\_{\mathbf{x}\in\mathcal{X}} | \operatorname{Q}(\mathbf{x}, \ \operatorname{Y})]]]\_{\mathbf{Y}=\operatorname{p}} \triangleq \operatorname{Q}[\mathbf{X}, \ \operatorname{\Phi}[\operatorname{Q}(\mathbf{X}, \ \operatorname{Y})]] \triangleq \operatorname{Q}[\mathbf{X}, \ \operatorname{\Phi}[\operatorname{Q}(\mathbf{N}\_{0}, \ \operatorname{Y})]] \triangleq \operatorname{Q} \end{aligned} \tag{4}$$

♦ Within the Gödel number/code *<sup>q</sup>*, *<sup>q</sup>* <sup>=</sup> *<sup>q</sup>* [17, 19], we perform the substitution *Y*: = *p* and then *X*: = *x* and write

<sup>6</sup> *Substitution function Sb* <sup>⋯</sup> � ⋯ ! is, in this way, similar to the computer *machine instruction* which itself,

is always able to realize its operation with its operands on the arbitrary storage place, but practically it is always applicated within the limited *address space* and within the given *operation regime/mode* of the computer's activity only (e.g. regime/mode *Supervisor* or *User*).

<sup>7</sup> Φ and *Z* represents the *Gödel numbering* and *Sb* the *Substitution*, *B*, *Bew* the *PA*-arithmetic *Proof*.

<sup>8</sup> Following the Gödel Proposition *V* (the first part) [5–7].

$$\begin{aligned} r[Z(\mathbf{x})] &= \text{Sb}\begin{pmatrix} \mathbf{17} & \mathbf{19} \\ q(\mathbf{17}, \mathbf{19}) \\ Z(\mathbf{x}) & Z(p) \end{pmatrix} = \text{Sb}\begin{pmatrix} \mathbf{17} \\ q[\mathbf{17}, Z(p)] \\ Z(\mathbf{x}) \end{pmatrix} \\ &= \Phi[\mathbf{Q}[\mathbf{x}, \ \Phi[\mathbf{Q}(\mathbf{X}, \ \mathbf{Y})]]] = \Phi[\mathbf{Q}[\mathbf{x}, \ \Phi[\mathbf{Q}(\mathbf{N\_0}, \ \mathbf{Y})]]] = \Phi[\mathbf{Q}(\mathbf{x}, \ \mathbf{p})] \end{aligned} \tag{5}$$

*<sup>P</sup>* of the *hyperplane* <sup>R</sup> *<sup>P</sup>*<sup>∈</sup> <sup>R</sup> ð Þ *xi <sup>n</sup>*

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

*integration factor for it*.

♣ **The states'** *<sup>θ</sup>*<sup>L</sup>

**Figure 1.**

**7**

*Adiabatic changes of the state of the system* L*, illustration.*

lOL are expressible **regularly:**

cannot reach at all, see the **Figure 1**.

*l*2*d, l*2*e, l*<sup>3</sup> is *QExt* = 0*,* Δ*QExt* = 0*,* d*QExt* = 0.

*i*¼1

*II*. **Caratheodory theorem** (() says that: ◊ *If the Pfaff form <sup>δ</sup><sup>Q</sup>* <sup>¼</sup> <sup>P</sup>*<sup>n</sup>*

*where Xi are continuously differentiable functions of n variables* (over a simply continuous area), has such a property that in the arbitrary vicinity of any arbitrarily

**such points which, from** *P*, **cannot be accessible along the path satisfying the equation** d*Q* = 0, then this form is **holonomous**; *it has or it is possible to find an*

**Caratheodory formulation** of the *II.* Law of Thermodynamics (⇔) claims that: ◊ *In the arbitrary vicinity of every state of the state space of the adiabatic system, there are such states that, from the given starting point, cannot be reached along an adiabatic path* (reversibly and irreversibly), or such states which the system

**Remark**: Now the symbol *Q* denotes that heat given to the state space of the thermodynamic system from its outside and directly; *Q* ≜ *QExt*; along paths *l*2*b, l*2*b*0*,*

½ �� **changes in the adiabatic system** L*=*OL, **along the trajectories**

*are inaccessible along the path satisfying the equation* d*Q* = 0.

*Common Gnoseological Meaning of Gödel and Caratheodory Theorems*

chosen and fixed point *<sup>P</sup>* of the hyperplane <sup>R</sup> *<sup>P</sup>*<sup>∈</sup> <sup>R</sup> ð Þ *xi <sup>n</sup>*

� � <sup>¼</sup> const*:* � �, *such points which*, *from this point P*,

*i*¼1

� � <sup>¼</sup> const*:* � �, **there exists**

*<sup>i</sup>*¼<sup>1</sup>*Xi*d*xi*,

With the great quantification of *r*[*Z*(*x*)] by *Z*(*x*) by the *VARIABLE X* (17), we have (similarly as in [4, 8]),

$$Z(\mathbf{x})Genr[Z(\mathbf{x})] = \mathbf{1}\mathsf{T}Gen\,q[\mathbf{1}\mathsf{T},\ \mathsf{Z}[\mathsf{1}\mathsf{T}Gen(\mathsf{1}\mathsf{T},\mathsf{1}\mathsf{9})]] = \mathbf{1}\mathsf{T}Gen\,r[\mathbf{1}\mathsf{T}] = \mathbf{1}\mathsf{T}Gen\,r\tag{6}$$

$$\triangleq \Phi[\forall\_{\mathbf{x}\in X} | \Phi[\mathbf{Q}[\mathbf{x},\Phi[\forall\_{\mathbf{x}\in X} | \mathbf{Q}(\mathbf{x},\mathbf{y})]]]] = \Delta Q[\mathbf{X},\Phi[\mathbf{Q}(\mathbf{X},\mathbf{y})]] = Q[\mathbb{N}\_{0},\Phi[\mathbf{Q}(\mathbb{N}\_{0},\mathbf{y})]] \tag{7}$$

#### **2.2 Gödel theorems**

*I*. **Gödel theorem** (corrected semantically by [3, 9, 10]) claims that

♣ **for every** *recursive* **and** *consistent CLASS OF FORMULAE <sup>κ</sup>* **and outside this set there is such true ("1")** *CLAIM r* **with free** *VARIABLE v r*≜*r v*ð Þ � � **that neither** *PROPOSITION vGen r* **nor** *PROPOSITION Neg*(*vGen r*) *belongs* **to the set** *Flg*(*κ*),

$$[vGen\,r \notin Flg(\kappa)] \quad \& \quad [\text{Neg}\,(vGen\,\,r) \notin Flg(\kappa)] \tag{7}$$

*FORMULA vGen r* **and** *Neg*(*vGen r*) **are not** *κ*-*PROVABLE—FORMULA vGen <sup>r</sup> is not <sup>κ</sup>*-*DECIDABLE*. They both are elements of inconsistent (meta)system <sup>P</sup><sup>∗</sup> .

*II*. Gödel **theorem** (corrected semantically according to [3, 9, 10]) claims that

♣ **if** *<sup>κ</sup>* **is an arbitrary** *recursive* **and** *consistent CLASS OF FORMULAE***, then any** *CLAIM* **saying that** *CLASS κ* **is consistent must be constructed outside this set, and for this fact it is not** *κ*-*PROVABLE.*


♦ The fact that the recursive *CLASS OF FORMULAE <sup>κ</sup>* (now *PA*—*Peano Arithmetic* especially) is consistent, is tested by *unary relation Wid*(*κ*), (die *Widerspruchsfreiheit*, *Consistency*) [5–7],

$$\operatorname{Wid}(\kappa) \sim (\operatorname{Ex}) \left[ \operatorname{Form}(\kappa) \quad \& \; \overline{\operatorname{Bew}\_{\kappa}(\kappa)} \right] \tag{8}$$


#### **3. Caratheodory theorems**

*<sup>I</sup>*. **Caratheodory's theorem** ()) says that: ◊ If the *Pfaff form has an integration factor, then there are, in the arbitrary vicinity of any arbitrarily chosen and fixed* point

<sup>9</sup> Far from (!) "In … ." in [5–7]

<sup>10</sup> Far from " … [*PA*-]arithmetic and sentencial/*SENTENCIAL*" in [5–7].

<sup>11</sup> Any attempt to prove/*TO PROVE* it (to infer/to *TO INFER* it) in the system <sup>P</sup>*<sup>κ</sup>* assumes or leads to the requirement for inconsistency of the consistent (!) system P*<sup>κ</sup>* (in fact we are entering into the inconsistent metasystem <sup>P</sup><sup>∗</sup> - see the real sense [4, 9] of the Proposition *<sup>V</sup>* in [5–7]).

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

*<sup>P</sup>* of the *hyperplane* <sup>R</sup> *<sup>P</sup>*<sup>∈</sup> <sup>R</sup> ð Þ *xi <sup>n</sup> i*¼1 � � <sup>¼</sup> const*:* � �, *such points which*, *from this point P*, *are inaccessible along the path satisfying the equation* d*Q* = 0.

*II*. **Caratheodory theorem** (() says that: ◊ *If the Pfaff form <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*, *where Xi are continuously differentiable functions of n variables* (over a simply continuous area), has such a property that in the arbitrary vicinity of any arbitrarily chosen and fixed point *<sup>P</sup>* of the hyperplane <sup>R</sup> *<sup>P</sup>*<sup>∈</sup> <sup>R</sup> ð Þ *xi <sup>n</sup> i*¼1 � � <sup>¼</sup> const*:* � �, **there exists such points which, from** *P*, **cannot be accessible along the path satisfying the equation** d*Q* = 0, then this form is **holonomous**; *it has or it is possible to find an integration factor for it*.

**Caratheodory formulation** of the *II.* Law of Thermodynamics (⇔) claims that: ◊ *In the arbitrary vicinity of every state of the state space of the adiabatic system, there are such states that, from the given starting point, cannot be reached along an adiabatic path* (reversibly and irreversibly), or such states which the system cannot reach at all, see the **Figure 1**.

**Remark**: Now the symbol *Q* denotes that heat given to the state space of the thermodynamic system from its outside and directly; *Q* ≜ *QExt*; along paths *l*2*b, l*2*b*0*, l*2*d, l*2*e, l*<sup>3</sup> is *QExt* = 0*,* Δ*QExt* = 0*,* d*QExt* = 0.

♣ **The states'** *<sup>θ</sup>*<sup>L</sup> ½ �� **changes in the adiabatic system** L*=*OL, **along the trajectories** lOL are expressible **regularly:**

**Figure 1.** *Adiabatic changes of the state of the system* L*, illustration.*

**Through the state space of** *FORMULAE* **of the system** P**, we "travel"similarly** by the inference rules, *Modus Ponens* especially [performed by a **Turing Machine** *TM***,** the inference of which is considerable as realized by the *information transfer process* within a **Shannon Transfer Chain X**ð Þ , K, **Y** described thermodynamically by a **Carnot Machine** *CM*].

**by permitted** (adiabatic, d*QExt* = 0) **changes** *l*2*b*, *l*2*b*0, *l*2*d*, *l*2*<sup>e</sup>* and *l*3, **inaccessible**. And certainly, thermodynamic states □ beyond these scales, within the hierarchically higher systems, are not accessible from the inside of the system L*=*T itself,

the state *θ*<sup>4</sup> from the state *θ*<sup>1</sup> along any simple path *l*2*b*, *l*2*b*0, *l*2*d*, *l*2*<sup>e</sup>* in the state

d*QExt* = 0, which means **under the opposite requirement** d*QExt* 6¼ 0, **it is possible to design or to construct** a (nonadiabatic) **path linking a certain point/state of the** *state space* OL **located, e.g., on** *l*2*<sup>e</sup>* **with the point/state** *θ***4**; **for example, it is the path** *l*<sup>4</sup> **from** *θ***<sup>1</sup> to** *θ***4**, now in a certain nonadiabatic system N, N ⫅ T where, from the view of possibilities of changes of the state, see **Figure 1**, is

□ <sup>∉</sup> OL*<sup>=</sup>*T, f g □ "OL*<sup>=</sup>*T; ◊ <sup>∉</sup> *<sup>l</sup>*2*<sup>b</sup>*, *<sup>l</sup>*2*b*<sup>0</sup>, *<sup>l</sup>*2*<sup>d</sup>*, *<sup>l</sup>*2*<sup>e</sup>* f g , *<sup>l</sup>*<sup>3</sup> , ◊

. Then the path *l*<sup>4</sup> in the state space OL<sup>0</sup>

*energetic relations* (ℰ), it is possible, see the **Figure 1**, to write,

E, <sup>N</sup> is implemented in <sup>L</sup><sup>0</sup>

<sup>E</sup> , <sup>N</sup><sup>þ</sup> is implemented in v <sup>L</sup><sup>∗</sup> , …

♣ **The states from the sets** OL � *<sup>l</sup>*<sup>Q</sup><sup>L</sup> f g, OL<sup>0</sup> � *<sup>l</sup>*QL

**forming, within the hierarchy of the systems** L, L<sup>0</sup>

**.** Without violation of the adiabacity of the system L, it is not possible to reach

♣ However, **outside the adiabacity of the system** <sup>L</sup> expressed by the relation

OL <sup>¼</sup> ON <sup>¼</sup> OT <sup>≜</sup> OL*<sup>=</sup>*<sup>T</sup> <sup>T</sup> <sup>⫆</sup> <sup>N</sup> <sup>⫌</sup> <sup>L</sup> (9)

. Further, it is possible to create for this nonadiabatic system N an alternative adiabatic system L<sup>0</sup> DL ð Þ <sup>0</sup> ⫆ DL enabling adiabatic-isochoric changes, e.g., *θ*2*<sup>e</sup>* ! *θ*4. .. Both the new adiabatic system L<sup>0</sup> and its nonadiabatic "model" N can be a subsystem of another but also adiabatic and imminently superior system L<sup>þ</sup> having another/wider range of the state quantities than it was for the original systems L

*=*N will be, from the point of L<sup>0</sup> of the imminently superior adiabatic system Lþ, the adiabatic one—the system L<sup>0</sup> is already isolated in L<sup>þ</sup> and the system L<sup>þ</sup> itself is already created in a certain system L<sup>∗</sup> imminently superior to it, as an isolated/

**..** From the view of the *possibilities to change the state*, or from the view of the

We introduce a symbol *l*OL½ �� for adiabatic paths in the state spaces OL½ �� ,

*l*QL ≜ *l*2*<sup>b</sup>*, *l*2*b*<sup>0</sup>, *l*2*<sup>d</sup>*, *l*2*<sup>e</sup>* f g , *l*<sup>3</sup> , *l*QL<sup>0</sup> ≜ *l*2*<sup>b</sup>*, *l*2*b*<sup>0</sup>, *l*2*<sup>d</sup>*, *l*2*<sup>e</sup>*, *l*3, *l*2*<sup>e</sup>* � *l<sup>θ</sup>*2*e*,*θ*<sup>4</sup> f g , *l*<sup>5</sup> , *l*QLþ, *l*QL <sup>∗</sup> , ⋯

, … **in the view of adiabacity and specification of the system** L **are**

 <sup>∪</sup> f g □ **, which is in the framework of the system** <sup>L</sup> **inaccessible/ unachievable as a whole and also in any of its subset and member.** However, the <sup>L</sup>-*inaccessibility* (adiabatic inaccessibility, especially of {◊} in the state space OL*<sup>=</sup>*T) **also means existence of the paths** *l*QL **of the adiabatic system** L. In the sense of the domain of solution of its (the L's) state equations, **they cannot be part of the**

L ⫋ L<sup>0</sup> ⫋ L<sup>þ</sup> ⫋ L<sup>∗</sup> ⫋ L∗ ∗ … , N ⫋ N<sup>0</sup> ⫋ N<sup>þ</sup> ⫋ N<sup>∗</sup> ⫋ N∗ ∗ … (10)

, N<sup>þ</sup> ⫋ L<sup>∗</sup>

*<sup>=</sup>*<sup>N</sup> ⫋ OL<sup>þ</sup> , OLþ*=*N<sup>0</sup> ⫋ OL <sup>∗</sup> … .

, QL<sup>þ</sup> � *<sup>l</sup>*QL

⫋ OL*<sup>=</sup>*<sup>T</sup>

E, <sup>N</sup><sup>0</sup> is implemented in <sup>L</sup><sup>þ</sup>

,

, Lþ, L<sup>∗</sup> , … , **a certain set**

*<sup>=</sup>*<sup>N</sup> of the system

(11)

without its (not adiabatical) widening, either, see the **Figure 1.**

*Common Gnoseological Meaning of Gödel and Caratheodory Theorems*

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

space OL,

valid that

and N, OL ⫅ OL<sup>0</sup>

<sup>N</sup><sup>0</sup> <sup>⫋</sup> <sup>L</sup><sup>þ</sup>

*l*QL ⫋ *l*QL<sup>0</sup> ⫋ *l*QL<sup>þ</sup> ⫋ *l*QL <sup>∗</sup> ⫋ …

**functionality of** L (but mark it).

N<sup>þ</sup> ⫋ L<sup>∗</sup>

OL<sup>∗</sup> � *l*QL

<sup>L</sup> <sup>¼</sup> ◊

O<sup>∗</sup>

**9**

L0

*<sup>=</sup>*<sup>N</sup> ⫅ OL<sup>þ</sup>

adiabatic substitute for the system N<sup>0</sup> OL<sup>0</sup>

**The thermodynamic model for the consistent** P*=*T PA **inference**, from its axioms or formulas having been inferred so far, **is created by the Carnot Machine's activity, which models the inference**. **This whole Carnot Machine** *CM* **runs in the wider adiabatic system** L*=*DL **and, in fact, is, in this way, creating these states,** [the *TM*'s, ð Þ **X**, K, **Y** 's, configurations are then modeled by the states *θ*<sup>L</sup> *<sup>i</sup>* ∈ OL of the adiabatic L*=*DL with this modeling *CM* inside], see the **Figure 2**.

**The** L**'s initial imbalance starts the** *θ***<sup>L</sup>** ½ �� **s states' sequence on a trajectory** <sup>l</sup>OL and **is given by** the modeled

**These adiabatic trajectories** lOL now **represent the norm** of the **consistency** (and resultativity) of the P*=*T PA**-inference/computing process** expressible also in terms of the **information transfer/heat energy transformation.**

♣ **The adiabatic property of the thermodynamic system** <sup>L</sup> **is always created** over the given scales of its state quantities—over their scale for a certain "creating" original (and not adiabatic) system T, and **by its** *outerly specification* **or** *the design/ construction* by means of **heat/adiabatic isolation of the space** *Vmax* **of the original system** T **that the system** (L*=*T) **can occupy**, and **after the system** L **has been** (as the adiabatic isolated original system T) **designed and set in the starting state** *<sup>θ</sup>*1, see **Figure 1. The state** *<sup>θ</sup>***<sup>4</sup> is a state** ◊ **of the set of states** {◊}. **These states are those ones in the Figure 1,** which, although they are in the given scale of state quantities *U* and *V* of the state space OL of the system L considered,

*U* ∈ 〈*Umin*, *Umax*〉 and *V* ∈ 〈*Vmin*,*Vmax*〉, are within it [in (the state space OL of) L]

**Figure 2.** *The mutual describability of the CM,* ð Þ *X*, K, *Y and TM.*

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

**by permitted** (adiabatic, d*QExt* = 0) **changes** *l*2*b*, *l*2*b*0, *l*2*d*, *l*2*<sup>e</sup>* and *l*3, **inaccessible**. And certainly, thermodynamic states □ beyond these scales, within the hierarchically higher systems, are not accessible from the inside of the system L*=*T itself, without its (not adiabatical) widening, either, see the **Figure 1.**

**.** Without violation of the adiabacity of the system L, it is not possible to reach the state *θ*<sup>4</sup> from the state *θ*<sup>1</sup> along any simple path *l*2*b*, *l*2*b*0, *l*2*d*, *l*2*<sup>e</sup>* in the state space OL,

♣ However, **outside the adiabacity of the system** <sup>L</sup> expressed by the relation d*QExt* = 0, which means **under the opposite requirement** d*QExt* 6¼ 0, **it is possible to design or to construct** a (nonadiabatic) **path linking a certain point/state of the** *state space* OL **located, e.g., on** *l*2*<sup>e</sup>* **with the point/state** *θ***4**; **for example, it is the path** *l*<sup>4</sup> **from** *θ***<sup>1</sup> to** *θ***4**, now in a certain nonadiabatic system N, N ⫅ T where, from the view of possibilities of changes of the state, see **Figure 1**, is valid that

$$\begin{array}{ll} \square \not\subseteq \mathsf{O}\_{\mathsf{L}/\mathsf{T}}, \{\square\} \not\subseteq \mathsf{O}\_{\mathsf{L}/\mathsf{T}}; & \Diamond \not\equiv \{l\_{2b}, l\_{2b}, l\_{2d}, l\_{2e}, l\_{3}\}, & \{\diamond\} \not\subseteq \mathsf{O}\_{\mathsf{L}/\mathsf{T}}\\ \hearrow \mathcal{O}\_{\mathsf{L}} = \mathcal{O}\_{\mathsf{N}} = \mathcal{O}\_{\mathsf{L}} \triangleq \mathsf{O}\_{\mathsf{L}/\mathsf{T}} & \mathsf{L} \not\equiv \mathsf{M} \not\equiv \mathsf{T} \end{array} \tag{9}$$

. Further, it is possible to create for this nonadiabatic system N an alternative adiabatic system L<sup>0</sup> DL ð Þ <sup>0</sup> ⫆ DL enabling adiabatic-isochoric changes, e.g., *θ*2*<sup>e</sup>* ! *θ*4.

.. Both the new adiabatic system L<sup>0</sup> and its nonadiabatic "model" N can be a subsystem of another but also adiabatic and imminently superior system L<sup>þ</sup> having another/wider range of the state quantities than it was for the original systems L and N, OL ⫅ OL<sup>0</sup> *<sup>=</sup>*<sup>N</sup> ⫅ OL<sup>þ</sup> . Then the path *l*<sup>4</sup> in the state space OL<sup>0</sup> *<sup>=</sup>*<sup>N</sup> of the system L0 *=*N will be, from the point of L<sup>0</sup> of the imminently superior adiabatic system Lþ, the adiabatic one—the system L<sup>0</sup> is already isolated in L<sup>þ</sup> and the system L<sup>þ</sup> itself is already created in a certain system L<sup>∗</sup> imminently superior to it, as an isolated/ adiabatic substitute for the system N<sup>0</sup> OL<sup>0</sup> *<sup>=</sup>*<sup>N</sup> ⫋ OL<sup>þ</sup> , OLþ*=*N<sup>0</sup> ⫋ OL <sup>∗</sup> … .

**..** From the view of the *possibilities to change the state*, or from the view of the *energetic relations* (ℰ), it is possible, see the **Figure 1**, to write,

$$
\mathfrak{L} \not\subseteq \mathfrak{L}' \not\subseteq \mathfrak{L}^+ \not\subseteq \mathfrak{L}^\* \not\subseteq \mathfrak{L}^{\*\*} \dots, \quad \mathfrak{N} \not\subseteq \mathfrak{N}' \not\subseteq \mathfrak{N}^+ \not\subseteq \mathfrak{N}^{\*\*} \dots \tag{10}
$$

<sup>N</sup><sup>0</sup> <sup>⫋</sup> <sup>L</sup><sup>þ</sup> E, <sup>N</sup> is implemented in <sup>L</sup><sup>0</sup> , N<sup>þ</sup> ⫋ L<sup>∗</sup> E, <sup>N</sup><sup>0</sup> is implemented in <sup>L</sup><sup>þ</sup> N<sup>þ</sup> ⫋ L<sup>∗</sup> <sup>E</sup> , <sup>N</sup><sup>þ</sup> is implemented in v <sup>L</sup><sup>∗</sup> , …

We introduce a symbol *l*OL½ �� for adiabatic paths in the state spaces OL½ �� ,

*l*QL ≜ *l*2*<sup>b</sup>*, *l*2*b*<sup>0</sup>, *l*2*<sup>d</sup>*, *l*2*<sup>e</sup>* f g , *l*<sup>3</sup> , *l*QL<sup>0</sup> ≜ *l*2*<sup>b</sup>*, *l*2*b*<sup>0</sup>, *l*2*<sup>d</sup>*, *l*2*<sup>e</sup>*, *l*3, *l*2*<sup>e</sup>* � *l<sup>θ</sup>*2*e*,*θ*<sup>4</sup> f g , *l*<sup>5</sup> , *l*QLþ, *l*QL <sup>∗</sup> , ⋯ *l*QL ⫋ *l*QL<sup>0</sup> ⫋ *l*QL<sup>þ</sup> ⫋ *l*QL <sup>∗</sup> ⫋ …

(11)

♣ **The states from the sets** OL � *<sup>l</sup>*<sup>Q</sup><sup>L</sup> f g, OL<sup>0</sup> � *<sup>l</sup>*QL , QL<sup>þ</sup> � *<sup>l</sup>*QL , OL<sup>∗</sup> � *l*QL , … **in the view of adiabacity and specification of the system** L **are forming, within the hierarchy of the systems** L, L<sup>0</sup> , Lþ, L<sup>∗</sup> , … , **a certain set** O<sup>∗</sup> <sup>L</sup> <sup>¼</sup> ◊ <sup>∪</sup> f g □ **, which is in the framework of the system** <sup>L</sup> **inaccessible/ unachievable as a whole and also in any of its subset and member.** However, the <sup>L</sup>-*inaccessibility* (adiabatic inaccessibility, especially of {◊} in the state space OL*<sup>=</sup>*T) **also means existence of the paths** *l*QL **of the adiabatic system** L. In the sense of the domain of solution of its (the L's) state equations, **they cannot be part of the functionality of** L (but mark it).

### **4. Analogy between adiabacity and** *PA***-inference**

♣ Now **the states on the adiabatic paths** *<sup>l</sup>*QL (of changes of the state of the adiabatic system L) are considered to be the analogues of *PA*-**arithmetic claims/ claims of the** *Peano Arithmetic* **theory** T PA (formulated/inferred/proved in P),


adiabatic analogy of the higher consistent inferential system P<sup>0</sup> is by L<sup>0</sup> , P<sup>0</sup> ⫌ P, … *:* .. Then the given specific **adiabatic path** *l*2*b*, *l*2*b*0, *l*2*d*, *l*2*e*, *l*<sup>3</sup> is an analog of certain

*deducible thread x* ! *B xk* **of the claim** *xk* **of the theory** T PA, where

$$\begin{aligned} \overrightarrow{\mathbf{x}} \quad & B\boldsymbol{x}\_{k} = (\mathbf{x}\_{1}, \ \mathbf{x}\_{2}, \ \ldots \ \mathbf{a}\_{k-1}, \ \mathbf{x}\_{k}) B\boldsymbol{x}\_{k} = \text{"1"} \\ \boldsymbol{x}\_{1} &\in \{\text{AXIOMS}\}^{\mathcal{P}} \quad \text{and} \quad \boldsymbol{\varkappa}\_{1} \cong \boldsymbol{\theta}\_{1} \\ \boldsymbol{\varkappa}\_{1}, \ \boldsymbol{\varkappa}\_{2}, \ \ldots, \boldsymbol{\varkappa}\_{k-1}, \ \boldsymbol{\varkappa}\_{k} \in \mathcal{T}\_{\mathcal{P}A} \quad \text{and} \quad \boldsymbol{\varkappa}\_{k} \cong \boldsymbol{\theta} \in \{\boldsymbol{\theta}\_{2b}, \ \boldsymbol{\theta}\_{2b'}, \ \boldsymbol{\theta}\_{2d}, \ \boldsymbol{\theta}\_{2c}, \ \boldsymbol{\theta}\_{3}\} \\ \boldsymbol{\varkappa}\_{2}, \ \ldots, \boldsymbol{\varkappa}\_{k-1} &\ \stackrel{\cong}{\simeq} \quad \boldsymbol{\theta} \in \{\{\{l\_{2b} - \boldsymbol{\theta}\_{2b}\}, \ \{l\_{2b'} - \boldsymbol{\theta}\_{2b'}\}, \ \{l\_{2d} - \boldsymbol{\theta}\_{2d}\}, \end{aligned}$$

$$\{l\_{2\epsilon} - \theta\_{2\epsilon}\}, \quad \{l\_3 - \theta\_3\}\\
\{ } \\
\}$$

♣ The **states from the space** QL*<sup>=</sup>*<sup>T</sup> **of the system** <sup>L</sup>*=*<sup>T</sup> satisfying the range of values of the state quantities *p* ∈〈*pmin*, *pmax*〉, *V* ∈〈*Vmin*,*Vmax*〉, *T* ∈〈*Tmin*, *Tmax*〉*= U* ∈ 〈*Umin*, *Umax*〉), **which are inaccessible along any of the adiabatic paths from** *<sup>l</sup>*<sup>Q</sup><sup>L</sup> , that means they are the states ◊ from the difference OL*<sup>=</sup>*<sup>T</sup> � *<sup>l</sup>*QL � �, shortly said from f g T � L , are considered to be analogues of not *PA*-claims such as, e.g., the **Fermat's Last Theorem**. <sup>12</sup> So, they are **analogues of** *all-the-time* **true** ("**1**") **arithmetic but not**-*PA*-**arithmetic claims**. From the point of adiabacity of the system <sup>L</sup>, they (◊) are only some thermodynamic states of its "creating" system <sup>T</sup>, and they are from the common range of values of the state quantities for T and L. From the point of expressing possibilities it as always true

$$
\mathfrak{T}^{[\cdot]} \nsubseteq \mathfrak{U}^{[\cdot]} \subseteq \mathfrak{T}^{[\cdot]} \nsubseteq \mathfrak{T}^{\star} \nsubseteq \left\{ \mathfrak{T}^{\mathbb{C}\_{[\cdot]}/\mathbb{C}\_{[\cdot]}} \right\}^{\star} \tag{13}
$$

(12)

**these systems** L, L<sup>0</sup>

the analog O<sup>∗</sup>

<sup>R</sup> *<sup>P</sup>*<sup>∈</sup> <sup>R</sup>, <sup>R</sup> ð Þ *xi <sup>n</sup>*

metasystem <sup>P</sup><sup>∗</sup> .

code 17*Gen r*".

namics language,

the *Circulus Viciosus*.

**11**

13

*l*QL , *l*QL<sup>0</sup> , *l*QLþ, *l*QL <sup>∗</sup> , … **running in them**.

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

QL*<sup>=</sup>*<sup>T</sup>

*i*¼1

means of the Thermodynamics language,

<sup>∃</sup>j*l*<sup>Q</sup><sup>L</sup> ð Þ) ð Þ) <sup>∃</sup>jQL <sup>∃</sup>jQ<sup>∗</sup>

QL � *l*QL

, Lþ, L<sup>∗</sup> , … ; **they confirm adiabacity of changes**

� � ⫋ Q<sup>∗</sup>

� �⊉*l*QL

� � ) <sup>∃</sup>jQL ð Þ) <sup>∗</sup> <sup>∃</sup><sup>j</sup> QL*<sup>=</sup>*<sup>T</sup>

� �<sup>⋆</sup>

; it as valid that

(15)

<sup>L</sup> ⫋ QL*<sup>=</sup>*<sup>T</sup> � �<sup>⋆</sup>

*<sup>i</sup>*¼<sup>1</sup>*Xi*d*xi*, *where Xi are functions of n vari-*

<sup>L</sup> "QL"*l*QL (17)

*<sup>i</sup>*¼<sup>1</sup>*Xi*d*xi has a integration factor*, *then there*

� � <sup>¼</sup> const*:* � � *along the path satis-*

h i (18)

� �<sup>⋆</sup> � � (16)

For (to illustrate our analogy) a supposedly countable set of states along the paths *l*QL of changes of the state of the system L (for simplicity we can consider the *isentrop l*2*<sup>e</sup>* only), the *PROPOSITION* 17*Gen r* is a claim of countability set nature,

*ables, continuously differentiable* (over a simply continuous domain) *has such a quality that in the arbitrary vicinity of arbitrarily chosen fixed point P of the hyperplane*

� � <sup>¼</sup> *<sup>C</sup>* <sup>¼</sup> const � *.*] *there exists* a set of points *inaccessible from the point P along the path satisfying the equation* d*QExt* = 0, *then it is possible to find an integration factor for it and then this form is holonomous*. In a physical sense and, by

**it says what**, in its consequence [*w* 17*Gen r*, (8)] and in a meta-arithmeticlogical way, the *II.* **Gödel theorem** (corrected semantically by [3, 9, 10]) **claims**; ♣ if *<sup>κ</sup>* is an arbitrary *recursive* and *consistent CLASS OF FORMULAE*, then any *CLAIM* (written as the *SENTENCIAL* and as such, representing a countable set of claims, which are its implementations) saying that *CLASS κ* is consistent must be

*UNPROVABLE* or cannot be *κ*-*PROVABLE*. In fact, it is a part of the inconsistent

"QL<sup>∗</sup> "Q<sup>∗</sup>

*are in the arbitrary vicinity of an arbitrarily chosen fixed point P of the hyperplane* R

*fying the equation* d*QExt* = 0. In a physical sense and by means of the Thermody-

� �<sup>⋆</sup> � <sup>Q</sup><sup>∗</sup>

<sup>13</sup> Any attempt to prove/*TO PROVE* it (to infer/*TO INFER* it) within the system <sup>P</sup>*<sup>k</sup>* assumes or leads to

*i*¼1

L � �"*l*<sup>Q</sup><sup>L</sup>


L

constructed outside this set and for this fact it is not *κ*-*PROVABLE*/is *κ*-

**.** In a physical sense and by the Thermodynamics language,

QL*<sup>=</sup>*<sup>T</sup> � �<sup>⋆</sup>

♣ It is possible to claim that, *I.* **Caratheodory theorem**,

◊ *if an arbitrary Pfaff form <sup>δ</sup>QExt* <sup>¼</sup> <sup>P</sup>*<sup>n</sup>*

*some points inaccessible from this point P P*<sup>∈</sup> <sup>R</sup> ð Þ *xi <sup>n</sup>*

Q<sup>∗</sup> <sup>L</sup> "*l*QL � �∧ QL*<sup>=</sup>*<sup>T</sup>

<sup>L</sup> of which is formulated in the set QL*<sup>=</sup>*<sup>T</sup>

� �<sup>⋆</sup> <sup>⫌</sup> QL and QL � *<sup>l</sup>*QL

*Common Gnoseological Meaning of Gödel and Caratheodory Theorems*

� �<sup>⊉</sup> QL and QL � *<sup>l</sup>*QL

**4.1 Analogy between Caratheodory and Gödel theorems**

We claim that, *II.* **Caratheodory theorem**, ◊ *if an arbitrary Pfaff form <sup>δ</sup>QExt* <sup>¼</sup> <sup>P</sup>*<sup>n</sup>*

[Symbol <sup>T</sup><sup>⋆</sup> denotes thermodynamic theory as a whole and symbol QL½ �� *<sup>=</sup>*T½ �� n o<sup>⋆</sup> is a mark for a transitive and reflexive closure of the set of (any) claims about systems L½ �� *==*T½ �� *:*].

♣ **The whole set** <sup>O</sup><sup>∗</sup> <sup>L</sup> **of states inaccessible in a given scale of state quantities** of the system L*=*T along the arbitrary **adiabatic path from** *l*<sup>Q</sup><sup>L</sup> in **the system** L (states ◊), as well as **the set of** <sup>L</sup>**-inaccessible states** □ **outside this scale**, see **Figure 1, are considered now to be the thermodynamic bearer of analogy of the semantics of the Gödel's** *UNDECIDABLE PROPOSITION* 17*Gen r*,

$$\begin{aligned} \mathfrak{17Gen} & \cong \mathfrak{Q}\_{\mathfrak{L}}^{\*} \quad \left[ = \{ \{\diamond\} \cup \{\square\} \} \right] \\ \mathfrak{Q}\_{\mathfrak{L}}^{\*} &= \{ \mathfrak{Q}\_{\mathfrak{L}^{\*}} \cdot l\_{\mathfrak{Q}\_{\mathfrak{L}}} \}, \quad \mathfrak{Q}\_{\mathfrak{L}}^{\*} \not\subseteq \mathfrak{Q}\_{\mathfrak{L}^{\*}} \quad \mathfrak{F}\_{\mathfrak{L}} \left\{ \mathfrak{Q}\_{\mathfrak{L}^{\*} \times \backslash \mathfrak{L}^{\*}} \right\}^{\star} \end{aligned} \tag{14}$$


<sup>12</sup> Alternatively Goldbach's conjecture.

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

**these systems** L, L<sup>0</sup> , Lþ, L<sup>∗</sup> , … ; **they confirm adiabacity of changes** *l*QL , *l*QL<sup>0</sup> , *l*QLþ, *l*QL <sup>∗</sup> , … **running in them**.

For (to illustrate our analogy) a supposedly countable set of states along the paths *l*QL of changes of the state of the system L (for simplicity we can consider the *isentrop l*2*<sup>e</sup>* only), the *PROPOSITION* 17*Gen r* is a claim of countability set nature, the analog O<sup>∗</sup> <sup>L</sup> of which is formulated in the set QL*<sup>=</sup>*<sup>T</sup> � �<sup>⋆</sup> ; it as valid that

$$\begin{array}{cccc}\left\{\mathfrak{Q}\_{\mathfrak{L}/\mathfrak{L}}\right\}^{\star} \nrightarrow{\mathfrak{Q}} \mathfrak{Q}\_{\mathfrak{L}} & \text{and} & \left\{\mathfrak{Q}\_{\mathfrak{L}} - l\_{\mathfrak{Q}\_{\mathfrak{L}}}\right\} \nrightarrow{\mathfrak{Q}} \mathfrak{Q}\_{\mathfrak{L}}^{\star} \notin \left\{\mathfrak{Q}\_{\mathfrak{L}/\mathfrak{L}}\right\}^{\star} \\\left\{\mathfrak{Q}\_{\mathfrak{L}} - l\_{\mathfrak{Q}\_{\mathfrak{L}}}\right\} \not\supseteq \mathfrak{Q}\_{\mathfrak{L}} & \text{and} & \left\{\mathfrak{Q}\_{\mathfrak{L}} - l\_{\mathfrak{Q}\_{\mathfrak{L}}}\right\} \not\models l\_{\mathfrak{Q}\_{\mathfrak{L}}} \end{array} \tag{15}$$

#### **4.1 Analogy between Caratheodory and Gödel theorems**

We claim that, *II.* **Caratheodory theorem**,

◊ *if an arbitrary Pfaff form <sup>δ</sup>QExt* <sup>¼</sup> <sup>P</sup>*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup>*Xi*d*xi*, *where Xi are functions of n variables, continuously differentiable* (over a simply continuous domain) *has such a quality that in the arbitrary vicinity of arbitrarily chosen fixed point P of the hyperplane* <sup>R</sup> *<sup>P</sup>*<sup>∈</sup> <sup>R</sup>, <sup>R</sup> ð Þ *xi <sup>n</sup> i*¼1 � � <sup>¼</sup> *<sup>C</sup>* <sup>¼</sup> const � *.*] *there exists* a set of points *inaccessible from the point P along the path satisfying the equation* d*QExt* = 0, *then it is possible to find an integration factor for it and then this form is holonomous*. In a physical sense and, by means of the Thermodynamics language,

$$(\exists |l\_{\mathfrak{U}}) \implies (\exists |\mathfrak{U}\_{\mathfrak{L}}) \implies (\exists |\mathfrak{U}\_{\mathfrak{L}}^{\*}) \implies (\exists |\mathfrak{U}\_{\mathfrak{L}^{\*}}) \implies \left(\exists |\{\mathfrak{U}\_{\mathfrak{L}^{\uparrow}\mathfrak{L}}\}^{\star}\right) \tag{16}$$

**it says what**, in its consequence [*w* 17*Gen r*, (8)] and in a meta-arithmeticlogical way, the *II.* **Gödel theorem** (corrected semantically by [3, 9, 10]) **claims**;

♣ if *<sup>κ</sup>* is an arbitrary *recursive* and *consistent CLASS OF FORMULAE*, then any *CLAIM* (written as the *SENTENCIAL* and as such, representing a countable set of claims, which are its implementations) saying that *CLASS κ* is consistent must be constructed outside this set and for this fact it is not *κ*-*PROVABLE*/is *κ*-*UNPROVABLE* or cannot be *κ*-*PROVABLE*. In fact, it is a part of the inconsistent metasystem <sup>P</sup><sup>∗</sup> .


**.** In a physical sense and by the Thermodynamics language,

$$\{\mathfrak{Q}\_{\mathbb{Z}/\mathbb{Z}}\}^{\star} \mkern-12014pt}{\mathfrak{Q}\_{\mathbb{Z}/\mathbb{Z}}\mathfrak{Q}\_{\mathbb{Z}}\mathfrak{q}\_{\mathbb{Z}} \mkern-12014pt} \mathfrak{Q}\_{\mathbb{Z}/\mathbb{Z}} \mkern-12014pt} \tag{17}$$

♣ It is possible to claim that, *I.* **Caratheodory theorem**,

◊ *if an arbitrary Pfaff form <sup>δ</sup>QExt* <sup>¼</sup> <sup>P</sup>*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup>*Xi*d*xi has a integration factor*, *then there are in the arbitrary vicinity of an arbitrarily chosen fixed point P of the hyperplane* R *some points inaccessible from this point P P*<sup>∈</sup> <sup>R</sup> ð Þ *xi <sup>n</sup> i*¼1 � � <sup>¼</sup> const*:* � � *along the path satisfying the equation* d*QExt* = 0. In a physical sense and by means of the Thermodynamics language,

$$\left[ \left[ \mathfrak{Q}\_{\mathfrak{Z}}^{\*} \overleftarrow{\mathfrak{Q}} \mathfrak{I}\_{\mathfrak{Q}\_{\mathbb{Z}}} \right] \land \left[ \left( \left\{ \mathfrak{Q}\_{\mathfrak{Z}/\mathbb{Z}} \right\}^{\star} - \mathfrak{Q}\_{\mathfrak{Z}}^{\*} \right) \overleftarrow{\mathfrak{Q}} \mathfrak{I}\_{\mathfrak{Q}\_{\mathbb{Z}}} \right] \tag{18}$$

<sup>13</sup> Any attempt to prove/*TO PROVE* it (to infer/*TO INFER* it) within the system <sup>P</sup>*<sup>k</sup>* assumes or leads to the *Circulus Viciosus*.

**it says what**, in a meta-arithmetic-logical way, the *I.* **Gödel theorem** (corrected semantically by [3, 9, 10]) **claims**;

♣ for every *recursive* and *consistent CLASS OF FORMULAE <sup>κ</sup>* and outside this set, there is such a true ("**1**") *CLAIM r* with free *VARIABLE v r*≜*r v*ð Þ � � that neither *PROPOSITION vGen r* nor *PROPOSITION Neg*(*vGen r) belongs* to the set *Flg*(*κ*),

½ � *vGen r* ∉ *Flg*ð Þ*κ* & ½ � *Neg vGen r* ð Þ ∉ *Flg*ð Þ*κ* (19)


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

and with **Figure 1**, we write for L*=*P

*l*QL ⊬ QL <sup>∗</sup> � *l*QL

*l*QL ⊬ *l*QL ⊬ QL <sup>∗</sup> � *l*QL

*<sup>y</sup>* <sup>¼</sup> *<sup>q</sup>*½ �ffi 17, 19 *<sup>θ</sup>*½ �� <sup>⊬</sup> ◊

*l*QL ⊬ *l*QL ⊬ *l*QL ⊬ QL <sup>∗</sup> � *l*QL

*<sup>θ</sup>*<sup>4</sup> <sup>∈</sup> QL<sup>∗</sup> � *<sup>l</sup>*<sup>Q</sup><sup>L</sup> f g <sup>¼</sup> ◊

*<sup>l</sup>*QL ffi 17, ◊

*<sup>θ</sup>*½ �� <sup>⊬</sup> *<sup>θ</sup>*½ �� <sup>⊬</sup> ◊

neatly

**13**

∀*<sup>θ</sup>*½ �� <sup>∈</sup>*l*QL

*<sup>θ</sup>*½ �� <sup>⊬</sup> *<sup>θ</sup>*½ �� <sup>⊬</sup> *<sup>θ</sup>*½ �� <sup>⊬</sup> ◊

fact, methodological axiom which has been formulated in a certain hierarchically higher inferential (meta)system <sup>P</sup><sup>∗</sup> , <sup>P</sup><sup>∗</sup> ffi f g OL <sup>⋆</sup>. In accordance with the above

� � <sup>∪</sup> f g □ � � <sup>¼</sup> <sup>Q</sup><sup>∗</sup>

� � � � � � <sup>∈</sup> f g QL <sup>∗</sup> <sup>⋆</sup> � *<sup>l</sup>*QL

� � � � � � � � <sup>∈</sup> f g QL <sup>∗</sup> <sup>⋆</sup> � *<sup>l</sup>*QL

and further, for the theory *l*OL *=*T PA, following (1)–(6) and [4], we write

� � <sup>∪</sup> f g □ � � ffi 19 19<sup>∈</sup> QL f g<sup>∗</sup> <sup>⋆</sup> � *<sup>l</sup>*QL

� � <sup>∪</sup> f g □ � � h i h i


♣ **The notation** <sup>17</sup>*Gen r* itself expresses the property of the system <sup>P</sup> and also the theory T PA, just as an *subject* which itself is not and cannot be the object of its own, and thus its notation **is not and cannot be one of the objects of the system** P

<sup>L</sup>½ �� ffi 17*Gen r* for L*=*P *vGen r* for L<sup>0</sup>

<sup>L</sup> "QL"*l*QL ].

½ �) *l*QL ⊬ *θ*<sup>4</sup> *l*QL ⊬ QL <sup>∗</sup> � *l*QL

*Common Gnoseological Meaning of Gödel and Caratheodory Theorems*

� � � � ffi *<sup>l</sup>*QL " QL <sup>∗</sup> � *<sup>l</sup>*QL

*l*QL ffi T *<sup>P</sup>*A, *card l*QL ¼ *card* T PA ¼ ℵ<sup>0</sup> *card*f g QL <sup>∗</sup> <sup>⋆</sup> <sup>¼</sup> <sup>ℵ</sup>1, *card* f g QL <sup>∗</sup> <sup>⋆</sup> � *<sup>l</sup>*QL

*yZ y* ½ �¼ ð Þ *<sup>q</sup>*½ �ffi 17, *<sup>y</sup> <sup>θ</sup>*½ �� <sup>⊬</sup> *<sup>θ</sup>*½ �� <sup>⊬</sup> ◊

� � <sup>∪</sup> f g □ � � � � � � <sup>∈</sup> ◊

� � <sup>∪</sup> f g □ � � � � � � � � , …

For 19: = *<sup>Z</sup>*(*p*) is *<sup>p</sup>*[*Z*(*p*)] = *<sup>r</sup>*(17) and *<sup>r</sup>*ð Þffi <sup>17</sup> ◊

h i ffi <sup>17</sup>*Gen*, <sup>∀</sup>*<sup>θ</sup>*½ �� <sup>∈</sup>*l*QL <sup>j</sup> *<sup>θ</sup>*½ �� <sup>⊬</sup> ◊

½ �ffi <sup>17</sup>*Gen r* <sup>∀</sup>*<sup>θ</sup>*½ �� <sup>∈</sup>*l*QL <sup>j</sup>*θ*½ �� ⊬ ∀*<sup>θ</sup>*½ �� <sup>∈</sup>*l*QL <sup>j</sup>*θ*½ �� <sup>⊬</sup> ◊

� � � � � � , …

½ �ffi 17*Gen r l*QL ⊬ *l*QL ⊬ OL<sup>∗</sup> � *l*QL

**accordance with Caratheodory we claim** that

which is the same as (21).

Q<sup>∗</sup>

[similarly, as (17) is valid, O<sup>∗</sup>

*<sup>θ</sup>*½ �� <sup>⊬</sup> ð Þ *<sup>θ</sup>* <sup>∪</sup> □ � �∈f g f g*<sup>θ</sup>* <sup>∪</sup> f g □ <sup>⋆</sup> � �

<sup>L</sup>½ �� is in the position of the analog for this, in

� � <sup>⫋</sup> f g QL <sup>⋆</sup> (21)

� � � �

� � � � , …

� � <sup>∪</sup> f g □ � �<sup>⋆</sup> and so we can write

(22)

(23)

<sup>L</sup> <sup>⫋</sup> QL f g<sup>∗</sup> <sup>⋆</sup> � *<sup>l</sup>*QL

� � � �

� � � �

� � <sup>∪</sup> f g □ � �<sup>⋆</sup> h i

� � <sup>∪</sup> f g □ � � � � h i ffi ½ � <sup>17</sup>*Gen q*½ � ð Þ 17, 19

*=*P<sup>0</sup> , L<sup>þ</sup> ½ � , … (24)

� � � � <sup>∈</sup> f g QL <sup>∗</sup> <sup>⋆</sup> � *<sup>l</sup>*QL

� � � �

� � <sup>¼</sup> <sup>1</sup>

� � <sup>∪</sup> f g □ � � � � , *<sup>p</sup>* <sup>¼</sup> <sup>17</sup>*Gen q*½ � 17, 19

� � <sup>∪</sup> f g □ � � � � � �

*FORMULA vGen r* and *Neg*(*vGen r*) are not *κ*-*PROVABLE—FORMULA vGen r i s not <sup>κ</sup>*-*DECIDABLE*. They are elements of inconsistent (meta)system <sup>P</sup> <sup>∗</sup> .

♣ **For us, as an isolated system** <sup>L</sup>, **to achieve such a "state**," it is necessary to consider the states with values of state quantities which are not a part of the domain of solution of the state equation for L. The system L has not been designed for them (so, we are facing inconsistency). For example, **the required volume** *V* **and temperature** *T* **should be greater than their maxima** *Vmax* and *Tmax* achievable by the system L. In order "to achieve" them, **the system** L **itself would have to "get out of itself,"and in order to obtain values** *V* **and** *T* **greater** than *Vmax* and *Tmax*, **it would have to "redesign"/reconstruct itself**. However, it is **us, being in a position of the hierarchically higher object**, who has to do so, from the outside the state space QL*<sup>=</sup>*<sup>T</sup> (from the outside the volume *Vmax*), which the system may occupy now.14


♣ The states unachievable within the state spaces of the systems L, L<sup>0</sup> , Lþ, L<sup>∗</sup> , … or inaccessible from them are creating, as a whole, a *certain class of equivalence or macrostate* O<sup>∗</sup> <sup>L</sup>½ �� <sup>Q</sup><sup>∗</sup> <sup>L</sup>½ �� "QL½ �� "*l*QL½ �� � � in hierarchy of the state spaces, from the point of their possible development, of always superior systems — <sup>L</sup><sup>0</sup> <sup>∪</sup> <sup>L</sup><sup>þ</sup> <sup>∪</sup> <sup>L</sup><sup>∗</sup> f g for <sup>L</sup>, <sup>L</sup><sup>þ</sup> <sup>∪</sup> <sup>L</sup><sup>∗</sup> f g for <sup>L</sup><sup>0</sup> , L<sup>∗</sup> for Lþ, *…* . The existence of the macrostate O<sup>∗</sup> <sup>L</sup>½ �� , already beginning from the original system <sup>L</sup> (macrostate <sup>O</sup><sup>∗</sup> <sup>L</sup> ), confirms the existence of the currently considered (adiabatic) system L½ �� and its properties, especially its adiabacity. And by this, in our analogy, it also con rms the consistency of its arithmetic/mathematical analog P,P<sup>0</sup> ,P<sup>þ</sup>, … (a complement of the set cannot exist without this set) and, on the contrary,

$$\begin{array}{rclclcl} \left[ \exists \left( \begin{matrix} \mathfrak{T} \big{(} \mathfrak{T}^{\ast/\mathsf{z}} \big{)} \star \end{matrix} \right) & \Rightarrow & \left( \begin{matrix} \exists \big{|\mathfrak{T}^{\ast}| \big{)} \end{matrix} \right) & \Rightarrow & \left( \begin{matrix} \exists \big{|\mathfrak{T}^{\ast/\mathsf{z}}| \big{)} \end{matrix} \right) \\ \left[ \left[ \begin{matrix} \left( \exists \big{|\mathfrak{T}^{\ast/\mathsf{z}}| \right) \right) \end{matrix} \right] & \Rightarrow & \left( \begin{matrix} \left( \left( \mathfrak{T} \big{)} \mathfrak{T}^{\ast/\mathsf{z}} \right) \end{matrix} \right) \end{array} \right] & \begin{matrix} \left( \begin{matrix} \left( \mathfrak{T} \big{)} \mathfrak{T}^{\ast/\mathsf{z}} \right) \end{matrix} \right) \\ \Rightarrow & \left( \begin{matrix} \left( \mathfrak{T} \big{)} \mathfrak{T}^{\ast/\mathsf{z}} \right) \end{matrix} \right) \\ \Rightarrow & \left[ \left( \begin{matrix} \left( \mathfrak{T} \big{)} \mathfrak{T}^{\ast}\_{\mathfrak{T}^{\ast}} \right) \end{matrix} \right) \end{array} \right] \end{array} \tag{20}$$


The specific states accessible in the state space OL ¼ f*p*∈〈*pmin*, *pmax*〉, *V* ∈〈*Vmin*,*Vmax*〉, *T* ∈〈*Tmin*, *Tmax*〉*=U* ∈〈*Umin*, *Umax*〉, … of the isolated system L through reversible or irreversible changes other than adiabatic are thermodynamic analogy (interpretation) of the enlargement of the axiomatics of the original system P½ � *<sup>κ</sup>* to the new system P<sup>0</sup> ,P<sup>þ</sup>, … , *similar/relative* to the P½ � *<sup>κ</sup>* . Such an enlargement of the system P to a certain system P½ � �� enabled Andrew Wiles to prove the Fermat's Last Theorem. Through its representative *θ*<sup>0</sup> we enlarge L to L<sup>0</sup> , L<sup>0</sup> ffi P<sup>0</sup> .

<sup>14</sup> This also involves introduction of the representative *θ*<sup>0</sup> of Fermat's Last Theorem provided we are speaking about L with *l*OL and provided we require enlargement L<sup>0</sup> in order to get L<sup>0</sup> ffi P<sup>0</sup> .

$$\mathfrak{G}\_{4} \in \{\mathfrak{Q}\_{\mathcal{L}} \colon l\_{\Omega\_{\mathbb{L}}}\} = \left\{ \left\{ \begin{matrix} \mathbb{G} \end{matrix} \right\} \cup \left\{ \begin{matrix} \square \end{matrix} \right\} \right\} = \mathfrak{Q}\_{\mathcal{L}}^{\*} \nsubseteq \left\{ \left\{ \begin{matrix} \Omega\_{\mathcal{L}^{\*}} \end{matrix} \right\}^{\star} - l\_{\Omega\_{\mathcal{L}}} \right\} \in \left\{ \left\{ \Omega\_{\mathcal{L}} \right\}^{\star} \right\} \qquad \text{(21)} \\\ \left[ l\_{\Omega\_{\mathcal{L}}} \nvdash\mathfrak{G}\_{\mathbb{L}} \right] \Rightarrow \left[ l\_{\Omega\_{\mathcal{L}}} \nvdash\left\{ \Omega\_{\mathcal{L}^{\*}} \ast - l\_{\Omega\_{\mathcal{L}}} \right\} \right] \quad \left[ \in \left\{ \left\{ \Omega\_{\mathcal{L}^{\*}} \right\}^{\star} - l\_{\Omega\_{\mathcal{L}}} \right\} \right] \end{matrix} \qquad \text{(21)} $$

$$\left[ l\_{\Omega\_{\mathcal{L}}} \nvdash\left\{ \Omega\_{\mathcal{L}^{\*}} \ast - l\_{\Omega\_{\mathcal{L}}} \right\} \right] \cong \left[ l\_{\Omega\_{\mathcal{L}}} \underline{\mathcal{L}} \left\{ \left\{ \Omega\_{\mathcal{L}^{\*}} \ast - l\_{\Omega\_{\mathcal{L}}} \right\} \right\} \right] \qquad \text{(22)} \\\ \left[ l\_{\Omega\_{\mathcal{L}}} \nvdash\left\{ l\_{\Omega\_{\mathcal{L}}} \nvdash\left\{ \Omega\_{\mathcal{L}^{\*}} \ast - l\_{\Omega\_{\mathcal{L}}} \right\} \right\} \right] \quad \left[ \in \left\{ \left\{ \left$$

$$\begin{aligned} \left[\Psi\_{\theta\_{\parallel} \in l\_{\Omega\_{\mathcal{E}}}}\right] & \cong \textsf{17Gen}, \quad \left[\Psi\_{\theta\_{\parallel} \in l\_{\Omega\_{\mathcal{E}}}}\left[|\theta\_{\parallel}\rangle \nvdash \{\{\diamond\}\cup\{\square\}\}\right]\right] \cong \left[\textsf{17Gen}\left[q\left(\textsf{17},\textsf{19}\right)\right]\right] \\\\ \left[\textsf{17Gen}\right] & \cong \left[\textsf{4}\_{\left[\raisebox{0.5pt}{ $\Omega\_{\parallel} \in l\_{\Omega\_{\mathcal{E}}}$ } $}\left|\theta\_{\parallel}\right|\nvdash \left[\textsf{4}\_{\left[\raisebox{0.5pt}{$ \Omega\_{\parallel} \in l\_{\Omega\_{\mathcal{E}}} $}$ }\left|\theta\_{\parallel}\right|\nvdash \{\left\{\raisebox{0.5pt}{ $\Omega\_{\parallel}$ }\right\}\dots\right|\nvdash \{\raisebox{0.5pt}{ $\Omega\_{\parallel}$ }\right]\right]\right] \\\\ \left[\textsf{17Gen}\right] & \cong \left[\textsf{17Gen}\right] \cong \left[\textsf{1\_{\Omega\_{\mathcal{E}}}}\varleftarrow\left[\textsf{1\_{\Omega\_{\mathcal{E}}}}\nrightarrow\left\{\textsf{2\_{\mathcal{E}}}\text{.}\left.-\textsf{l}\_{\Omega\_{\mathcal{E}}}\right\}\right]\right], \ldots \end{aligned} \tag{23}$$

$$
\mathfrak{U}^{\*}\_{\mathfrak{V}^{\natural}, \mathbb{L}} \cong \mathfrak{U} \mathsf{Gen} \ r \quad \text{for} \ \mathfrak{L}/\mathcal{P} \quad [v \mathsf{Gen} \ r \quad \text{for} \ \mathfrak{L}'/\mathcal{P}', \ \mathfrak{L}^{+}, \dots] \tag{24}
$$

**Demonstration:** Following (8) ½ � *Wid*ð Þ) P 17*Gen r* , we claim for the systems L*=*P that

**5. Conclusion**

totality.

**transfer**).

**15**

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

*Common Gnoseological Meaning of Gödel and Caratheodory Theorems*

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

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**

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)].

**the recursive and axiomatizable system can never be proved in it itself**,

Now, **with our better comprehension**, we can claim that the **consistency of**

d*Q*<sup>L</sup> *Ext* <sup>¼</sup> <sup>0</sup> ffi *<sup>w</sup>*, d*Q*<sup>L</sup> *Ext* <sup>¼</sup> <sup>0</sup> � <sup>L</sup>; *<sup>w</sup>* ffi <sup>d</sup>*Q*<sup>L</sup> *Ext* <sup>¼</sup> <sup>0</sup> , <sup>L</sup> � <sup>d</sup>*Q*<sup>L</sup> *Ext* <sup>¼</sup> <sup>0</sup> ð Þ) <sup>∃</sup>j<sup>L</sup> <sup>∃</sup>jO<sup>∗</sup> L , <sup>∃</sup>jQ<sup>∗</sup> L ffi <sup>17</sup>*Gen r*; <sup>∃</sup>jO<sup>∗</sup> L ) ð Þ <sup>∃</sup>j<sup>L</sup> ,ð Þffi <sup>∃</sup>j<sup>L</sup> <sup>17</sup>*Gen r* ð Þ) <sup>∃</sup>j<sup>L</sup> <sup>∃</sup>jO<sup>∗</sup> L ffi ½ � *<sup>w</sup>* ) ð Þ <sup>17</sup>*Gen r* ; <sup>∃</sup>jO<sup>∗</sup> L ) ð Þ <sup>∃</sup>j<sup>L</sup> ffi ½ � ð Þ) <sup>17</sup>*Gen r <sup>w</sup>* so, that <sup>∃</sup>jO<sup>∗</sup> L ) ð Þ <sup>∃</sup>j<sup>L</sup> ffi ½ � ð Þ) <sup>17</sup>*Gen r <sup>w</sup>* & ð Þ) <sup>∃</sup>j<sup>L</sup> <sup>∃</sup>jO<sup>∗</sup> L ffi ½ � *<sup>w</sup>* ) ð Þ <sup>17</sup>*Gen r* and then <sup>∃</sup>jO<sup>∗</sup> L � <sup>17</sup> *Gen r* (25)

*I.* Gödel theorem (corrected semantically by [3, 9, 10]):

**For every** *recursive* **and consistent** *CLASS OF FORMULAE κ*, and **outside this set,** there **exists the true ("1")** *CLAIM* **r with a free** *VARIABLE* **v that neither the** *CLAIM* **vGen r nor the** *CLAIM* **Neg(vGen r)** *belongs* **to the set Flg**(*κ*)

½ � *vGen r* ∉ *Flg*ð Þ*κ* & ½ � *Neg vGen r* ð Þ ∉ *=Flg*ð Þ*κ* ,

*CLAIMS* **vGen r and Neg(vGen r) are not** *κ*-*PROVABLE***,** the *CLAIM* **vGen r** *is not κ*-*DECIDABLE***.**

[They are elements of the formulating/syntactic metasystem *κ*<sup>⋆</sup>, inconsistent against *κ*].

**II***.* **Gödel theorem** (corrected semantically by [3, 9, 10]):

**If** *κ* **is an arbitrary** *recursive* **and** *consistent CLASS OF FORMULAE,* **then any** *CLAIM* **saying that** *CLASS κ* **is consistent must be constructed outside** this set and for this fact, it **is not** *κ*-*PROVABLE***.**

The **consistency** of the *CLASS OF FORMULAE κ* is**tested** by the *relation***Wid**(*κ*).

$$\operatorname{Wid}(\kappa) \sim (\operatorname{Ex})[\operatorname{CLAIM}(\mathbf{x}) \& \; \overline{\operatorname{Proof}\_{\kappa}(\mathbf{x})}]$$

**The** FORMULAE **class** *κ* **is consistent.**

⇔

*at least one κ*-*UNPROVABLE CLAIM* **x exists.**

**Now x = 17Gen r** <sup>∉</sup> <sup>P</sup>*=*<sup>T</sup> PA, *<sup>κ</sup>* ¼ T PA, <sup>T</sup> PA <sup>⊂</sup><sup>P</sup> <sup>⊂</sup>P<sup>⋆</sup>

Then, **semantically understood** and with the language of logic and metaarithmetics, the full meaning of the **Gödel proof expresses the universal validity of the** *II.* **Law of Thermodynamics**. 15

$$[II.\text{P.T.}[\text{Proofof}\_{\mathcal{P}}(\text{17Goon.}) = \text{"0"}] = \text{"1"} \ ] = \text{Wid}(\mathcal{T}\_{\mathcal{P}.4})$$

<sup>15</sup> Our consideration is based on the similarity between the Cantor diagonal argument used in construction of the Gödel Undecidable Formula and the proof way of the Caratheodory theorems; adiabacity/consistency is prooved by leaving them and sustaining their validity - paradox.
