Open access peer-reviewed chapter

Common Gnoseological Meaning of Gödel and Caratheodory Theorems

By Bohdan Hejna

Submitted: February 5th 2019Reviewed: June 11th 2019Published: October 30th 2019

DOI: 10.5772/intechopen.87975

Downloaded: 298


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


  • 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 spaceof an adiabatic thermodynamic systemwith arithmetic formulasand the Peano inferenceis compared with the adiabatic changeswithin this state space. The whole set of the statesnow not achievable adiabaticallyrepresents 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 butin 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 inferenceis 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 formulationof 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 SystemP/TheoryTPA.

♣ “1″ - arithmeticity of the Padiabacity of the L/OL.

♣ Consistent TPAinference within Pmoving along trajectories 1DLin DL/L.

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

Remark: Any inferencewithin the system P2 sets the TPA-theoreticalrelation3 among its formulae a. This relation is given by their gradually generated special sequenceaa1aqapakak+1, which is the proofof the latest inferred formula ak + 1. By this, the uniquearithmetic relation between their Gödel numbers, FORMULAE x[·], x[·] = Φ(a[·]), is set up, too. The gradually arising SEQUENCE of FORMULAEx=Φais the PROOFof its latest FORMULA xk + 1.

Let us assume that the given sequence a=ao1ao2aoaqapakak+1is a special one, and that, except of axioms (axiomatic schemes) a01, , ao, it has been generated by the correct application of the rule Modus Ponens only.4

Within the process of the (Gödelian) arithmetic-syntactic analysisof the latest formula ak + 1 of the proof a, we use, from the aselected, (special) subsequence aq,p,k+1of the formulae aq, ap, ak + 1. The formulae aq, aphave already been derived, or they are axioms. It is valid that q, p < k + 1, and we assume that q < p,


Checking the syntactic andTPA-theoretical correctnessof the analyzed chains ai, as the formulae of the system Phaving been generated by inferring (Modus Ponens) within the system P(in the theory TPA), and also the special sequence of the formulae aof the system (theory TPA), is realized by checking the arithmetic-syntacticcorrectness of the notation of their corresponding FORMULAEand 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 Sbu1u2tZxZyor the expression Sb1719tZxZyrepresents the result value of the Gödel number t[Z(x), Z(y)], which is coding the (constant) claim T(x, y) z PMhas been generated by the substitution of xa yinstead of the free variables Xand Yin the function T(X, Y) from PMwith its Gödelian code t(u1, u2) in the (arithmetized) P,


Into theVARIABLES,we substitute theSIGNSof the sametypebut the introduction of the termadmissible substitutionitself 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 notFORMULAEof the systemP(and thus not belonging into the theoryTPA).

Then the substitution functionSbis not possible, within the frame of the inference in the systemP, 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 languageLPand, above all, within the frame of the conditions specified by the právě a jenomINFERENCEof the elements of the languageLTPAonly (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 Ppossible.6

2.1 The Gödel UNDECIDABLE CLAIM’s construction

♦ Let the Gödel numbers xand ybe given. The number xis the SEQUENCE OF FORMULAEvalid and yis a FORMULAof P. We define the valid constant relation Q(x, y) from the Q(X, Y) for given values xand y, X:=x, Y:= y; 17 = Φ(X), 19 = Φ(Y),78


♦ Now we put p = 17Gen q, q = q(17, 19) q1719QXYand then,


The meta-language symbol QXYor QN0Yis to be read: NoxXN0is in theκ-INFERENCErelation to the variableY(to its space of values Y).

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

rSb19qZpand thenr=Sb19q1719Zp=r17=ΦQXpE3

The Gödel number ris, by the substitution of the NUMERAL Z(p), supposedly only(by [5, 6, 7]) the CLASS SIGNwith the FREE VARIABLE17 (X); with the values p,the rcontains the feature of autoreference,


♦ Within the Gödel number/code q, q = q[17, 19], we perform the substitution Y: = pand then X: = xand 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 everyrecursiveandconsistent CLASS OF FORMULAEκand outside this set there is such true (“1”)CLAIM rwith freeVARIABLEvrrvthat neitherPROPOSITIONvGen rnorPROPOSITIONNeg(vGen r) belongsto the setFlg(κ),


FORMULAvGen randNeg(vGen r) are notκ-PROVABLE—FORMULAvGen ris notκ-DECIDABLE. They both are elements of inconsistent (meta)system P.

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

ifκis an arbitraryrecursiveandconsistent CLASS OF FORMULAE, then anyCLAIMsaying thatCLASS κis consistent must be constructed outside this set, and for this fact it is notκ-PROVABLE.

- Outside9 the consistent systemPκ, there is a true(“1”) formula,10 the ARITHMETIZATIONof which isκ-UNPROVABLE FORMULA17Gen r.11

♦ The fact that the recursive CLASS OF FORMULAE κ(now PAPeano Arithmeticespecially) is consistent, is tested by unaryrelationWid(κ), (die Widerspruchsfreiheit, Consistency) [5, 6, 7],


- a class ofFORMULAEκis consistentDefthere existsat least oneFORMULAx[PROPOSITION x(x = 17Gen r)], which isκ-UNPROVABLE.

3. Caratheodory theorems

I. Caratheodory’s theorem(⇒) says that: ◊ If the Pfaff formhas an integration factor,then there are, in the arbitrary vicinity of any arbitrarily chosen and fixedpoint Pof the hyperplaneRPRxii=1n=const., such points which, from this point P, are inaccessiblealong the path satisfying the equationdQ = 0.

II. Caratheodory theorem(⇐) says that: ◊ If thePfaff formδQ=i=1nXidxi, 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 Pof the hyperplane RPRxii=1n=const., there exists such points which, fromP, cannot be accessible along the path satisfying the equationdQ = 0, then this form is holonomous; it has or it is possible to find an integration factor for it.

Caratheodory formulationof 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.

Figure 1.

Adiabatic changes of the state of the systemL, illustration.

Remark: Now the symbol Qdenotes that heat given to the state space of the thermodynamic system from its outside and directly; QQExt; along paths l2b, l2b′, l2d, l2e, l3 is QExt = 0,QExt = 0,dQExt = 0.

The states’θLchanges in the adiabatic systemL/OL, along the trajectorieslOLare expressible regularly:

Through the state space ofFORMULAEof the systemP, we “travel” similarlyby the inference rules, Modus Ponensespecially [performed by a Turing MachineTM,the inference of which is considerable as realized by the information transfer processwithin a Shannon Transfer ChainXKYdescribed thermodynamically by a Carnot MachineCM].

The thermodynamic model for the consistentP/TPAinference, 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 MachineCMruns in the wider adiabatic systemL/DLand, in fact, is, in this way, creating these states,[the TM’s, XKY‘s, configurations are then modeled by the states θiLOLof the adiabatic L/DLwith this modeling CMinside], see the Figure 2.

Figure 2.

The mutual describability of theCM,XKYandTM.

TheL‘s initial imbalance starts theθLs states’ sequence on a trajectorylOLand is given bythe modeled

These adiabatic trajectorieslOLnow represent the normof the consistency(and resultativity) of the P/TPA-inference/computing processexpressible also in terms of the information transfer/heat energy transformation.

The adiabatic property of the thermodynamic systemLis always createdover the given scales of its state quantities—over their scale for a certain “creating” original (and not adiabatic) system T, and by itsouterly specificationorthe design/constructionby means of heat/adiabatic isolation of the spaceVmaxof the original systemTthat the system(L/T) can occupy, and after the systemLhas been(as the adiabatic isolated original system T) designed and set in the starting stateθ1, see Figure 1. The stateθ4is a stateof the set of states{◊}. These states are those ones in theFigure 1,which, although they are in the given scale of state quantities Uand Vof the state space OLof the system Lconsidered, UUminUmaxand VVminVmax, are within it [in (the state space OLof) L] by permitted(adiabatic, dQExt = 0) changesl2b, l2b′, l2d, l2eand l3, inaccessible. And certainly, thermodynamic states □ beyond these scales, within the hierarchically higher systems, are not accessible from the inside of the system L/Titself, 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 θ4 from the state θ1 along any simple path l2b, l2b′, l2d, l2ein the state space OL,

♣ However, outside the adiabacity of the systemLexpressed by the relation dQExt = 0, which means under the opposite requirementdQExt0, it is possible to design or to constructa (nonadiabatic) path linking a certain point/state of thestate spaceOLlocated, e.g., onl2ewith the point/stateθ4; for example, it is the pathl4 fromθ1toθ4, now in a certain nonadiabatic system N, NTwhere, from the view of possibilities of changes of the state, see Figure 1, is valid that


. Further, it is possible to create for this nonadiabatic system Nan alternative adiabatic system LDLDLenabling adiabatic-isochoric changes, e.g., θ2e → θ4.

.. Both the new adiabatic system Land its nonadiabatic “model” Ncan be a subsystem of another but also adiabatic and imminently superior system L+having another/wider range of the state quantities than it was for the original systems Land N, OLOL/NOL+. Then the path l4 in the state space OL/Nof the system L/Nwill be, from the point of Lof the imminently superior adiabatic system L+, the adiabatic one—the system Lis already isolated in L+and the system L+itself is already created in a certain system Limminently superior to it, as an isolated/adiabatic substitute for the system NOL/NOL+OL+/NOL.

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


NL+E, Nis implemented in L, N+LE, Nis implemented in L+N+LE,N+is implemented in v L, …

We introduce a symbol lOL·for adiabatic paths in the state spaces OL·,


The states from the setsOLlQL,OL'lQL,QL+lQL,OLlQL,in the view of adiabacity and specification of the systemLare forming, within the hierarchy of the systemsL,L,L+,L,,a certain setOL=, which is in the framework of the systemLinaccessible/unachievable as a whole and also in any of its subset and member.However, the L-inaccessibility(adiabatic inaccessibility, especially of {◊} in the state space OL/T) also means existence of the pathslQLof the adiabatic systemL. In the sense of the domain of solution of its (the L‘s) state equations, they cannot be part of the functionality ofL(but mark it).

4. Analogy between adiabacity and PA-inference

♣ Now the states on the adiabatic pathslQL(of changes of the state of the adiabatic system L) are considered to be the analogues of PA-arithmetic claims/claims of thePeano ArithmetictheoryTPA(formulated/inferred/proved in P),

- adiabacity of the systemLis the analog of consistencyof the system Pκand

. the setlQLof adiabatic paths inOL/Tis an analog of PA-theoryTPA; then, adiabatic analogy of the higher consistent inferential system Pis by L, PP,.

.. Then the given specific adiabatic pathl2b, l2b′, l2d, l2e, l3 is an analog of certain deducible threadxBxkof the claimxkof the theoryTPA, where


♣ The states from the spaceQL/Tof the systemL/Tsatisfying the range of values of the state quantities ppminpmax, VVminVmax, TTminTmax/UUminUmax), which are inaccessible along any of the adiabatic paths fromlQL, that means they are the states ◊ from the difference OL/TlQL, shortly said from TL, are considered to be analogues of not PA-claims such as, e.g., the Fermat’s Last Theorem.12 So, they are analogues ofall-the-timetrue(“1”) arithmetic but not-PA-arithmetic claims. From the point of adiabacity of the system L, they (◊) are only some thermodynamic states of its “creating” system T, and they are from the common range of values of the state quantities for Tand L. From the point of expressing possibilities it as always true


[Symbol Tdenotes thermodynamic theory as a whole and symbol QL/Tis a mark for a transitive and reflexive closure of the set of (any) claims about systems L//T.].

The whole setOLof states inaccessible in a given scale of state quantitiesof the system L/Talong the arbitrary adiabatic path fromlQLin the systemL(states ◊), as well as the set ofL-inaccessible statesoutside this scale, see Figure 1, are considered now to be the thermodynamic bearer of analogy of the semantics of the Gödel’sUNDECIDABLE PROPOSITION17Gen r,


- The states fromOL(from QLlQL,QLlQL,QLlQL+,,QLlQL,,QLlQL,) inaccessible by permitted changes in currently used systemsL,L,L+,L,(within the scale of values of their state quantities and also out of this scale) confirm both existence and properties of these systemsL,L,L+,L,; they confirm adiabacity of changeslQL,lQL,lQL+,lQL,running in them.

For (to illustrate our analogy) a supposedly countable set of states along the paths lQLof changes of the state of the system L(for simplicity we can consider the isentrop l2eonly), the PROPOSITION17Gen ris a claim of countability set nature, the analog OLof which is formulated in the set QL/T; it as valid that


4.1 Analogy between Caratheodory and Gödel theorems

We claim that, II.Caratheodory theorem,

if an arbitraryPfaff formδQExt=i=1nXidxi, 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 hyperplaneR[PR,Rxii=1n=C=const.] there existsa set of points inaccessible from the point P along the path satisfying the equationdQExt = 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 [w17Gen r, (8)] and in a meta-arithmetic-logical way, the II.Gödel theorem(corrected semantically by [3, 9, 10]) claims;

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

- Outside the consistent system Pκ, there is a true (“1”) formula whose ARITHMETIZATIONis κ-UNPROVABLE FORMULA/PROPOSITION/CLAIMor code 17Gen r”.13

.In a physical sense and by the Thermodynamics language,


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

if an arbitraryPfaff formδQExt=i=1nXidxihas a integration factor, then there are in the arbitrary vicinity of an arbitrarily chosen fixed point P of the hyperplaneRsome points inaccessible from this pointPPRxii=1n=const.along the path satisfying the equationdQExt = 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 recursiveand consistent CLASS OF FORMULAE κand outside this set, there is such a true (“1”) CLAIM rwith free VARIABLEvrrvthat neither PROPOSITION vGen rnor PROPOSITION Neg(vGen r) belongsto the set Flg(κ),


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

For us, as an isolated systemL, 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 Lhas not been designed for them (so, we are facing inconsistency). For example, the required volumeVand temperatureTshould be greater than their maximaVmaxand Tmaxachievable by the system L. In order “to achieve” them, the systemLitself would have to “get out of itself,” and in order to obtain valuesVandTgreaterthan Vmaxand 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/T(from the outside the volume Vmax), which the system may occupy now.14

- This “procedure” corresponds to the CLAIM/PROPOSITION/FORMULA17Gen rconstruction by means of (Cantor’s) diagonal argumentand Caratheodory proof.

♣ The states unachievable within the state spaces of the systems L,L,L+,L,or inaccessible from them are creating, as a whole, a certain class of equivalence or macrostateOLQLQLlQLin hierarchy of the state spaces, from the point of their possible development, of always superior systems—LL+Lfor L,L+Lfor L,Lfor L+, . The existence of the macrostate OL, already beginning from the original system L(macrostate OL), confirms the existence of the currently considered (adiabatic) system Land its properties, especially its adiabacity. And by this, in our analogy, it also con rms the consistency of its arithmetic/mathematical analog P,P,P+,(a complement of the set cannot exist without this set) and, on the contrary,


- Based just upon this point of view, we assign the set/macrostate or equivalence class OLthe meaning of the bearer of the sense of the Gödel’sUNDECIDABLE PROPOSITION17Gen rforP,

- the L-unachievabilityof the set OLis in the position of the analog for this, in fact, methodological axiom which has been formulated in a certain hierarchically higher inferential (meta)system P,POL. In accordance with the above and with Figure 1, we write for L/P


and further, for the theory lOL/TPA, following (1)(6) and [4], we write


For 19: = Z(p) is p[Z(p)] = r(17) and r17and so we can write neatly


which is the same as (21).

- It is obvious from our thermodynamic analogy that CLAIM/PROPOSITION17Gen rfor has to be true and in connection with Gödel’sII.theorem, and in accordance with Caratheodory we claimthat


The notation17Gen ritself expresses the property of the system Pand also the theory TPA, just as an subjectwhich 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 systemP[similarly, as (17) is valid, OLQLlQL].

Demonstration:Following (8) WidP17Genr, we claim for the systems L/Pthat

dQExtL=0w,dQExtL=0L;wdQExtL=0,LdQExtL=0LOL,QL17Genr;OLL,L17GenrLOLw17Genr;OLL17Genrwso,thatOLL17Genrw&LOLw17Genrand thenOL17GenrE25

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

For everyrecursiveand consistentCLASS OF FORMULAEκ, and outside this set,there exists the true (“1”)CLAIMr with a freeVARIABLEv that neither theCLAIMvGen r nor theCLAIMNeg(vGen r)belongsto the set Flg(κ)


CLAIMSvGen r and Neg(vGen r) are notκ-PROVABLE,the CLAIMvGen ris notκ-DECIDABLE.

[They are elements of the formulating/syntactic metasystem κ, inconsistent against κ].

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

Ifκis an arbitraryrecursiveandconsistent CLASS OF FORMULAE,then anyCLAIMsaying thatCLASSκis consistent must be constructed outsidethis set and for this fact, it is notκ-PROVABLE.

The consistencyof the CLASS OF FORMULAEκis testedby the relationWid(κ).

TheFORMULAE classκis consistent.

at least oneκ-UNPROVABLE CLAIMx exists.

Now x = 17Gen rP/TPA, κ=TPA, TPAPP

Then, semantically understoodand with the language of logic and meta-arithmetics, the full meaning of the Gödel proof expresses the universal validity of theII.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 itselfjust 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 waythe 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 senseof 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 ofouter observation/measuring/studies, which is not possible within the system itself. Ignoring this approach is also the reason for the formulation of the Gibbs paradoxand Halting Problem. Also, our awareness of this fact results in our full understandingof 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 theII.Thermodynamic Principle.




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]QExt = 0, of the systemL,it is not possible to derivesuch aCLAIMthat is stating this adiabatic supposition.This CLAIMis constructible not adiabatically,outsidethe adiabaticLonly.

Under the consistency of the systemP,it is not possible to derivesuch aCLAIMthat is stating this consistency supposition.This CLAIMis constructiblepurely syntactically,outsidethe consistentPonly (inPP) (Figure A1).

Figure A1.

Example of not distinguishing the reality and its image.

WithoutPwe could not know that P is not self-referencing and is consistent.

A.2 The proof way of Caratheodory theorems

I. Let the form δQ=i=1nXixihas the integration factor vand let dR=i=1n1vXidxi. Then the Pfaff equation δQ=i=1nXidxi=0has the solution in the form Rx1xk=const. and this solution represents a family of hyperplanes in n-dimensional space, not intersecting each other. Let us pick now the pointPx10xn0determined by our choice ofconst. = C. Only the points lying in the hyperplaneRx10xn0are accessible from the pointPalong the path satisfying the conditiondQ = 0. All the points not lying in this hyperplane are inaccessible from the pointPalong the path satisfying the conditiondQ = 0 (Figure A2).

Figure A2.

The proof way of the Caratheodory theorems.

II. Let us pick the point V, e.g., from R3, lying in a vicinity of the point P, which is not accessible from Pfollowing the path dQ = 0. Let gbe a line going through the point Pand let gbe oriented (g) in such way that it does not satisfy the condition dQ = 0. The point Vand the line gdetermine a plane Xi = Xi(u, v), i = 1,2,3. Let us consider a curve kin this plane, going through the point V(u0, v0) in that way (g) that dQ = 0 is supposedly valid along this curve. There is only one curvekfor the pointV(u0, v0). It lies in our plane, the plane Xi = Xi(u, v), and then it is valid for it dXi=Xiudu+Xivdvand, considering dQ = 0 along k, we get i=13XiXiudu+i=13XiXivdv=0.

The curvek, however, intersects the linegin the pointR, which is inaccessible from the pointPalong the path withdQ = 0 (for dQRg0). Otherwise, the pointVwould also be accessible from the pointPthroughRandkdQRk=0, which is a conflict with the original assumption.By a suitable selection of V, it is possible to have the point Rarbitrarily close to the point P; in the arbitrary vicinity of the point P,there are points inaccessible from the point Palong the path with dQ = 0. Now, let us pick a line g′parallel to the line g, and a cylinder Cgoing through these two lines. We consider that the curve ksatisfying the relation dQ = 0 is on this cylinder C′goes through the point Pand intersects the line g′in the point M.

Now, let us consider another cylinder C′as the continuation of Cwith g′and g. Let us use the symbol k′for the continuation of the curve kin C′. Then the curve k′must intersect the line gin 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 Ninto the point Pand at the moments of discrepancy of the points Pand N, it would be possible to reach the point Pfrom the point Nalong the line g(supposedly with dQ = 0). However, the condition dQ = 0 is not valid there (dQRg0). By deforming C′into C, the kand k′would close a plane Fwhere dQ = 0. If the equation of this plane has the form Rxii=13=const., then the equation dQ = 0 has a solution—an integration factor for the Pfaff formδQ=i=13Xidxiexists [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]):

Figure A3.

The concept for ceasing the autoreference.

For ∆A″,it is valid in the cycle Othat


and, further, for ∆Ain the cycle O, we have


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


For the whole work ∆A* of the combined cycle OO, we have


Then, for the whole change of the thermodynamic entropy within the combined cycle OO(measured in information units 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 workafter the work ∆Ahas been performed at the temperature TW. Evidently, the sense of the symbol TW(within the double cycle OOand when ∆Q0 = ∆Q″0) is expressible by the symbol T* 0, which is possible, for the working temperatures of the whole cycle OOare TWand T″W = T*0. The relation (30) expresses that fact that the double cycle OOis the direct Carnot Cycle just with its working temperatures TW > T″W = T*0. In the double cycle OO, it is valid that


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


It is ensured by the propositions TW > T″W, T″0 = T0 and also by that fact that the loss entropy H(X|Y) is described and given by the heat ∆Q0 = ∆Q0. But in our combined cycle OO, it is valid too that


and we have


For the whole information entropy ΔAkTW(the whole thermodynamic entropy SCin information units) and by following the previous relations also it is valid that


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,


The cycles O,O, and OOare the Carnot Cycles, and thus from their definition and construction, they are imaginatively16 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 OOenables us to recognize that the infinite cycle Ois running. In our case, it is the infinite cycle from (5), (6) and also from [4, 8, 10],



  • 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.

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Bohdan Hejna (October 30th 2019). Common Gnoseological Meaning of Gödel and Caratheodory Theorems, Ontological Analyses in Science, Technology and Informatics, Andino Maseleno and Marini Othman, IntechOpen, DOI: 10.5772/intechopen.87975. Available from:

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