## Abstract

We will demonstrate that the I. and the II. Caratheodory theorems and their common formulation as the II. Law of Thermodynamics are physically analogous with the real sense of the Gödel’s wording of his I. and II. incompleteness theorems. By using physical terms of the adiabatic changes the Caratheodory theorems express the properties of the Peano Arithmetic inferential process (and even properties of any deductive and recursively axiomatic inference generally); as such, they set the physical and then logical limits of any real inference (of the sound, not paradoxical thinking), which can run only on a physical/thermodynamic basis having been compared with, or translated into the formulations of the Gödel’s proof, they represent the first historical and clear statement of gnoseological limitations of the deductive and recursively axiomatic inference and sound thinking generally. We show that semantically understood and with the language of logic and meta-arithmetics, the full meaning of the Gödel proof expresses the universal validity of the II. law of thermodynamics and that the Peano arithmetics is not self-referential and is consistent.1

### Keywords

- arithmetic formula
- thermodynamic state
- adiabatic change
- inference

## 1. Introduction

To show that the real/physical sense of the Gödel incompleteness theorems—that the very real sense of them—is the meta-arithmetic-logical analog of the Caratheodory’s claims about the *adiabatic system* (that they are the analog of the sense of the *II.* Law of Thermodynamics), we compare the states in the *state space* of an *adiabatic thermodynamic system* with *arithmetic formulas* and the *Peano inference* is compared with the *adiabatic changes* within this state space. The *whole set of the states* now *not achievable adiabatically* represents the existence of the states on an adiabatic path, but this fact is not expressible adiabatically. This property of which is the analog of the sense *of Gödel undecidable formula*. Nevertheless, any of these states, now not achievable adiabatically in the given state space (of the given adiabatic system), is achievable adiabatically *but* in the redefined and wider adiabatic system with its state space divided between adiabatic and not adiabatic parts again. These states (which are achievable only when the previous subsystem is part of the new actual system, both are consistent/adiabatic) represent arithmetic but not the Peano arithmetic formulas and also are bearing the property of their whole set. Also they can be axioms of the higher/superior inference including the previous one—the *general arithmetic inference* is further ruled by the same and repeated principle of widening the axiomatics and with same thermodynamic analogy using the redefined and widened new adiabatic system and its settings and with the same limitation by the impossibility to proof both the consistency of the given inferential system and, in our analogy, the adiabacity of its given adiabatic analog, by means of themselves. The consistency of the inferential system and adiabacity of its analog (and their abilities generally) are defined and proved by outer construction, outer limitations, and outer settings only (compare this our claim with the Gödel’s claim for the Peano arithmetic inference “… in the Peano arithmetic system exists …”).

**Caratheodory common formulation** of the *II*. **P.T.**:

In our considerations, we use the states of the adiabatic system as the thermodynamic representation of the Peano arithmetically inferred formulas and the transition between the stats is then the thermodynamic model of the Peano arithmetic inference step, the consistency of the Peano arithmetics is represented by the adiabacity of the modeling thermodynamic system.

**Peano Axioms/Inference Rules in the System** **/Theory**

♣ “1″ - arithmeticity of the

♣ Consistent

♣ The states on the adiabatic trajectories, also irreversible, then model the consistently inferred/inferrable *PA-FORMULAS*.

**Remark**: Any *inference* within the system ^{2} sets the *theoretical* relation^{3} among its formulae *special sequence* *proof* of the latest inferred formula *a*_{k + 1}. By this, the *unique* arithmetic relation between their *Gödel numbers*, *FORMULAE x*_{[·]}, *x*_{[·]} = Φ(*a*_{[·]}), is set up, too. The gradually arising *SEQUENCE of FORMULAE* *PROOF* of its latest *FORMULA x*_{k + 1}.

Let us assume that the given sequence *a*_{01}, *…*, *ao*, it has been generated by the correct application of the rule *Modus Ponens only*.^{4}

Within the process of the *(Gödelian) arithmetic-syntactic analysis* of the latest formula *a*_{k + 1} of the proof *selected*, (special) subsequence *aq, ap, a*_{k + 1}. The formulae *aq*, *ap* have already been derived, or they are axioms. It is valid that *q, p < k* + 1, and we assume that *q < p*,

Checking the *syntactic and* *theoretical correctness* of the analyzed chains *ai*, as the formulae of the system *Modus Ponens*) within the system *arithmetic-syntactic* correctness of the notation of their corresponding *FORMULAE* and *SEQUENCE of FORMULAE*, by means of the relations *Form*(·)*, FR*(·), *Op*(·,·,·), *Fl*(·,·,·) “called” from (the sequence of procedures) relations *Bew*(·)*,* (··)*B*(·)*, Bw*(·);^{5} the core of the whole (Gödelian) arithmetic-syntactic analysis is the (procedure) relation of *Divisibility*,

## 2. Gödel theorems

**Remark:** The expression *t*[*Z*(*x*)*, Z*(*y*)], 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)

♣ **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** **(and thus not belonging into the theory** **).**

**Then the substitution function** **is not possible, within the frame of the inference in the system** **, 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** **and, above all, within the frame of the conditions specified by the právě a jenom INFERENCE of the elements of the language**

Others than/semantically (or by the type) homogenous application of the substitution function is not within the right inference/INFERENCE within the system ^{6}

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

♦ Let the Gödel numbers *x* and *y* be given. The number *x* is the *SEQUENCE OF FORMULAE* valid and *y* is a *FORMULA* of *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}

♦ Now we put *p* = 17*Gen q*, *q* = *q*(17, 19)

The meta-language symbol **No** **is in the** *κ*-**INFERENCE relation to the variable** *Y* (to its space of values

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

The Gödel number *r* is, by the substitution of the *NUMERAL Z*(*p*), **supposedly only** (by [5, 6, 7]) the *CLASS SIGN* with the *FREE VARIABLE* 17 (*X*); with the values *p,* the *r* contains the feature of *autoreference*,

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

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

### 2.2 Gödel theorems

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

♣ **for every recursive and consistent CLASS OF FORMULAE** *κ* **and outside this set there is such true (“1”) CLAIM r with free VARIABLE** **that neither PROPOSITION** *vGen r* **nor PROPOSITION** *Neg*(*vGen r*) **belongs to the set** *Flg*(*κ*),

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

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

♣ **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.**

- Outside^{9} **the consistent system** **there is a true** (“**1**”) **formula**,^{10} the **ARITHMETIZATION of which is** *κ*-**UNPROVABLE FORMULA** 17*Gen r*.^{11}

♦ The fact that the recursive *CLASS OF FORMULAE κ* (now *PA*—*Peano Arithmetic* especially) is consistent, is tested by *unary relation Wid*(*κ*), (die *Widerspruchsfreiheit*, *Consistency*) [5, 6, 7],

- **a class of FORMULAE** *κ* **is consistent** **there exists at least one** *FORMULA x* [*PROPOSITION x* (*x* = 17*Gen r*)], **which is** *κ*-**UNPROVABLE**.

## 3. Caratheodory theorems

*I*. **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 *P* of the *hyperplane* **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** *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 *P* of the hyperplane **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; *l*_{2b}*, l*_{2b′}*, l*_{2d}*, l*_{2e}*, l*_{3} is *QExt* = 0*,* ∆*QExt* = 0*,* d*QExt* = 0.

♣ **The states’** **changes in the adiabatic system** **along the trajectories** **regularly:**

**Through the state space of FORMULAE of the system** **, 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** **Carnot Machine CM**].

**The thermodynamic model for the consistent** **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** **and, in fact, is, in this way, creating these states,** [the **TM**’s, **CM** inside], see the Figure 2.

**The** **‘s initial imbalance starts the** **s states’ sequence on a trajectory** **is given by** the modeled

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

♣ **The adiabatic property of the thermodynamic system** **is always created** over the given scales of its state quantities—over their scale for a certain “creating” original (and not adiabatic) system **by its outerly specification or the design/construction** by means of **heat/adiabatic isolation of the space** *Vmax* **of the original system** **that the system** (**can occupy**, and **after the system** **has been** (as the adiabatic isolated original system **designed and set in the starting state θ**_{1}, see Figure 1**. The state θ**_{4} **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 **by permitted** (adiabatic, d*QExt* = 0) **changes** *l*_{2b}, *l*_{2b′}, *l*_{2d}, *l*_{2e} 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 **.**

**.** Without violation of the adiabacity of the system **θ**_{4} from the state **θ**_{1} along any simple path *l*_{2b}, *l*_{2b′}, *l*_{2d}, *l*_{2e} in the state space

♣ However, **outside the adiabacity of the system** *QExt* = 0, which means **under the opposite requirement** **it is possible to design or to construct** a (nonadiabatic) **path linking a certain point/state of the state space** **located, e.g., on** *l*_{2e} **with the point/state θ**_{4}; **for example, it is the path** *l*_{4} **from θ**_{1} **to θ**_{4}, now in a certain nonadiabatic system

. Further, it is possible to create for this nonadiabatic system **θ**_{2e} → **θ**_{4}.

.. Both the new adiabatic system *l*_{4} in the state space

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

We introduce a symbol

♣ **The states from the sets** **in the view of adiabacity and specification of the system** **are forming, within the hierarchy of the systems** **a certain set** **, which is in the framework of the system** **inaccessible/unachievable as a whole and also in any of its subset and member.** However, the **inaccessibility** (adiabatic inaccessibility, especially of {◊} in the state space **also means existence of the paths** **of the adiabatic system** **they cannot be part of the functionality of**

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

♣ Now **the states on the adiabatic paths** *PA*-**arithmetic claims/claims of the Peano Arithmetic theory**

- **adiabacity of the system** **consistency** of the system

**. the set** **of adiabatic paths in** *PA*-**theory**

.. Then the given specific **adiabatic path** *l*_{2b}, *l*_{2b′}, *l*_{2d}, *l*_{2e}, *l*_{3} is an analog of certain **deducible thread** **of the claim** *xk* **of the theory**

♣ The **states from the space** **of the system** **which are inaccessible along any of the adiabatic paths from** *PA*-claims such as, e.g., the **Fermat’s Last Theorem**.^{12} So, they are **analogues of all-the-time true** (“**1**”) **arithmetic but not**-*PA*-**arithmetic claims**. From the point of adiabacity of the system

[Symbol

♣ **The whole set** **of states inaccessible in a given scale of state quantities** of the system **adiabatic path from** **the system** **the set of** **-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*,

- **The states from** **inaccessible by permitted changes in currently used systems** **confirm both existence and properties of these systems** **they confirm adiabacity of changes** **running in them**.

For (to illustrate our analogy) a supposedly countable set of states along the paths *isentrop l*_{2e} only), the *PROPOSITION* 17*Gen r* is a claim of countability set nature, the analog

### 4.1 Analogy between Caratheodory and Gödel theorems

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

◊ *if an arbitrary* **Pfaff form** *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* *.*] *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,

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

♣ if *κ* 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

- Outside the consistent system **1**”) formula whose *ARITHMETIZATION* is *κ*-*UNPROVABLE FORMULA/PROPOSITION/CLAIM* or code 17*Gen r*”.^{13}

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

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

◊ *if an arbitrary* **Pfaff form** *has a integration factor*, *then there are in the arbitrary vicinity of an arbitrarily chosen fixed point P of the hyperplane* *some points inaccessible from this point* *along the path satisfying the equation* d*QExt* = 0. In a physical sense and by means of the Thermodynamics language,

**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 κ* and outside this set, there is such a true (“**1**”) *CLAIM r* with free *VARIABLE* *PROPOSITION vGen r* nor *PROPOSITION Neg*(*vGen r) belongs* to the set *Flg*(*κ*),

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

♣ **For us, as an isolated system** **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 **the required volume** *V* **and temperature** *T* **should be greater than their maxima** *Vmax* and *Tmax* achievable by the system **the system** **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 *Vmax*), which the system may occupy now.^{14}

- This “procedure” corresponds to the **CLAIM/PROPOSITION/FORMULA** 17*Gen r* construction by means of (Cantor’s) **diagonal argument** and **Caratheodory proof**.

♣ The states unachievable within the state spaces of the systems *certain class of equivalence or macrostate* *…*. The existence of the macrostate

- Based just upon this point of view, we assign the set/macrostate or equivalence class **Gödel’s UNDECIDABLE PROPOSITION** 17*Gen r* **for**

- the *unachievability* of the set

and further, for the theory

For 19: = *Z*(*p*) is *p*[*Z*(*p*)] = *r*(17) and

which is the same as (21).

- It is obvious from our thermodynamic analogy that *CLAIM/PROPOSITION* 17*Gen r* for has to be true and **in connection with Gödel’s** *II.* **theorem, and in accordance with Caratheodory we claim** that

♣ **The notation** 17*Gen r* itself expresses the property of the system *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**

**Demonstration:** Following (8)

*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*)

**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*].

**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*).

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

⇔

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

**Now x = 17Gen r**

Then, **semantically understood** and with the language of logic and meta-arithmetics, the full meaning of the **Gödel proof expresses the universal validity of the** *II.* **Law of Thermodynamics**.^{15}

## 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**.

♣♦◊

## Acknowledgments

Supported by the grant of Ministry of Education of the Czech Republic MSM 6046137307.

Many thanks are to be expressed to my brother Ing. Petr Hejna for his help with English language and formulations of both this and all the previous texts.

## A. Appendix

## A.1 Summarizing comparison

♣ **Under the adiabacity**, [d]**Q**_{Ext} = **0**, **of the system** **, it is not possible to derive such a CLAIM that is stating this adiabatic supposition.** This **CLAIM** is **constructible not adiabatically, outside the adiabatic** **only.**

♣ **Under the consistency of the system** **, it is not possible to derive such a CLAIM that is stating this consistency supposition.** This **CLAIM** is **constructible purely syntactically, outside the consistent** **only (in** **) (**Figure A1**).**

♣ **Without** **we could not know that P is not self-referencing and is consistent.**

## A.2 The proof way of Caratheodory theorems

*I*. Let the form *v* and let *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 hyperplane** **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 *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 (*Q* = 0. The point *V* and the line *g* determine a plane *Xi* = *Xi*(*u, v*), *i* = 1*,* 2*,* 3. Let us consider a curve *k* in this plane, going through the point *V* (*u*_{0}*, v*_{0}) in that way (*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 *Q* = 0 along *k*, we get

**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 **Otherwise, the point** *V* **would also be accessible from the point** *P* **through** *R* **and** **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′* 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′* goes through the point *P* and intersects the line *g′* in the point *M*.

Now, let us consider another cylinder *C′* as the continuation of *C* with *g′* and *g*. Let us use the symbol *k′* for the continuation of the curve *k* in *C′*. Then the curve *k′* must intersect the line *g* in the point *P*. Otherwise, it would be possible to deform the plane *C′* 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 (*C′* into *C*, the *k* and *k′* would close a plane *F* where d*Q* = 0. If the equation of this plane has the form _{,} then the equation d*Q* = 0 has a solution—an integration factor for the **Pfaff form**

## 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 ∆*A″,* it is valid in the cycle

and, further, for ∆*A* in the cycle _{,} we have

and thus, for the cycles _{,} it is valid that

For the whole work ∆*A*^{*} of the combined cycle _{,} we have

Then, for the whole change of the thermodynamic entropy within the combined cycle *Hartley, nat, bit*) and thus for the change of the whole information entropy *H*^{*}(*Y*^{*}), it is valid that

It is valid, for ∆*A*^{*} is a *residuum work* after the work ∆*A* has been performed at the temperature *TW*. Evidently, the sense of the symbol *Q*_{0} = ∆*Q″*_{0}) is expressible by the symbol *T*^{*} _{0}, which is possible, for the working temperatures of the whole cycle *TW* and *T″W* = *T*^{*}_{0}. The relation (30) expresses that fact that the double cycle *TW > T″W* = *T*^{*}_{0}. In the double cycle _{,} it is valid that

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

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

and we have

For the whole information entropy

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

The cycles ^{16} in principle, the infinite cycles; in each of them the following *criterion of an infinite cycle* (see [12]) it is valid inevitably,

The construction of the cycle

## Notes

- The reader of the paper should be familiar with the Gödel proof’s way and terminology; SMALL CAPITALS in the whole text mean the Gödel numbers and working with them. This chapter is based, mainly, on the [1, 2, 3, 4]. This paper is the continuation of the lecture Gödel Proof, Information Transfer and Thermodynamics [4].
- Formal arithmetic inferential system.
- Peano Arithmetics Theory.
- For simplicity. The ‘real’ inference is applied to the formula ai + 1 for i = o.
- Formula, Reihe von Formeln, Operation, Folge, Glied, Beweis, Beweis, see Definition 1–46 in [5, 6, 7] and by means of all other, by them ‘called’, relations and functions (by their procedures).
- Substitution function Sb⋯⋅⋯ 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).
- Φ and Z represents the Gödel numbering and Sb the Substitution, B, Bew the PA-arithmetic Proof.
- Following the Gödel Proposition V (the first part) [5, 6, 7].
- Far from (!) “In ….” in [5, 6, 7]
- Far from “… [PA-]arithmetic and sentencial/SENTENCIAL” in [5, 6, 7].
- Any attempt to prove/TO PROVE it (to infer/to TO INFER it) in the system Pκ assumes or leads to the requirement for inconsistency of the consistent (!) system Pκ (in fact we are entering into the inconsistent metasystem P∗ - see the real sense [4, 9] of the Proposition V in [5, 6, 7]).
- Alternatively Goldbach’s conjecture.
- Any attempt to prove/TO PROVE it (to infer/TO INFER it) within the system Pk assumes or leads to the Circulus Viciosus.
- This also involves introduction of the representative θ0 of Fermat’s Last Theorem provided we are speaking about L with lOL and provided we require enlargement L′ in order to get L′≅P′. The specific states accessible in the state space OL={p∈pminpmax, V∈VminVmax, T∈TminTmax/U∈UminUmax,… 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 Pκ to the new system P′,P+,…, similar/relative to the Pκ. Such an enlargement of the system P to a certain system P⋅⋅ enabled Andrew Wiles to prove the Fermat’s Last Theorem. Through its representative θ0 we enlarge L to L′,L′≅P′.
- 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.
- When an infinite reserve of energy would exist.