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Technology » "Ontology in Information Science", book edited by Ciza Thomas, ISBN 978-953-51-3888-4, Print ISBN 978-953-51-3887-7, Published: March 8, 2018 under CC BY 3.0 license. © The Author(s).

Chapter 12

Information Transfer and Thermodynamic Point of View on Goedel Proof

By Bohdan Hejna
DOI: 10.5772/intechopen.68809

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Information Transfer and Thermodynamic Point of View on Goedel Proof

Bohdan Hejna
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Formula of an arithmetic theory based on Peano Arithmetics (including it) is a chain of symbols of its super‐language (in which the theory is formulated). Such a chain is in convenience both with the syntax of the super‐language and with the inferential rules of the theory (Modus Ponens, Generalization). Syntactic rules constructing formulas of the theory are not its inferential rules. Although the super‐language syntax is defined recursively—by the recursive writing of mathematical‐logical claims—only those recursively written super‐language’s chains which formulate mathematical‐logical claims about finite sets of individual of the theory, computable totally (thus recursive) and always true are the formulas of the theory. Formulas of the theory are not those claims which are true as for the individual of the theory, but not inferable within the theory (Great Fermat’s Theorem). They are provable but within another theory (with both Peano and further axioms). Also the chains expressing methodological claims, even being written recursively (Goedel Undecidable Formula) are not parts of the theory. The same applies to their negations. We show that the Goedel substitution function is not the total one and thus is not recursive. It is not defined for the Goedel Undecidable Formula’s construction. For this case, the structure of which is visible clearly, we are adding the zero value. This correction is based on information, thermodynamic and computing considerations, simplifies the Goedel original proof, and is valid for the consistent arithmetic theories directly.

Keywords: arithmetic formula, inference, information transfer, information entropy, heat efficiency, infinite cycle

1. Introduction

The formula of an arithmetic theory based on Peano Arithmetics (including it) is a chain of symbols of its metalanguage in which the theory is formulated such that it is both in convenience with the syntax of the metalanguage and with the inferential rules of the theory [of the inferential system (Modus Ponens, Generalization)].

Syntactic rules constructing formulae of the theory (but not only!) are not its inferential rules. Although the metalanguage syntax is defined recursively—by the recursive writing of mathematical‐logical claims, only those recursively written metalanguage’s chains which formulate mathematical‐logical claims about finite (precisely recursive) sets of individual of the theory, computable totally (thus recursive) and as always true are the formulae of the theory. Formulas of the theory are not those claims which are true as for the individual of the theory, but not inferable within the theory (Great Fermat’s Theorem). They are provable but within another theory (with further axioms than only those of Peano). Also the chains expressing methodological claims, even being written recursively (Goedel Undecidable Formula), are not parts of the theory, and also they are not parts of the inferential system; the same is for their negations.

We show that the Goedel substitution function is not the total one and thus is not recursive. It is not defined for the Goedel Undecidable Formula’s construction. For this construction, the structure of which is visible clearly, we are setting the zero value. This correction is based on information, thermodynamic and computing considerations, simplifies the Goedel original proof, and is valid for the consistent arithmetic theories directly.1

Remark: Paradoxical claims (paradoxes, noetical paradoxes, contradictions and antinomia) have two parts—both parts are true, but the truth of one part denies the truth of the second part.

They can arise by not respecting the metalanguage (semantic) level—which is the higher level of our thinking about problems and the language (syntactic) level—which is the lower level of formulations of our ‘higher’ thoughts. Also they arise by not respecting a double‐level organization and description of measuring—by not respecting the need of a ‘step‐aside’ of the observer from the observed. And also they arise by not respecting various time clicks in time sequences. As for the latter case, they are in a contradiction with the causality principle. The common feature for all these cases is the Auto‐Reference construction which itself, solved by itself, always states the requirement for ceasing the II. Principle of Thermodynamics and all its equivalents [10, 11, 12, 13].

Let us introduce the Russel’s criterion for removing paradoxes2: Within the flow of our thinking and speech we need and must distinguish between two levels of our thinking and expressing in order not to fall in a paradoxical claim by mutual mixing and changing them.

These levels are the higher one, the metalanguage (semantic) level and the lower one, the language (syntactic) level. Being aware of the existence of these two levels, we prevent ourselves from their mutual mixing and changing, we prevent ourselves from application our metalanguage claims on themselves but now on the language level or vice versa.

We must be aware that our claims about properties of considered objects are created on the higher level, rather richer both semantically and syntactically than the lower one on which we really express ourselves about these objects. The words and meanings of this lower (and ‘narrower’) level are common to both of them. Our speech is formulated and performed on the lower level describing here our ‘higher’ thoughts and on which the objects themselves have been described, defined yet too, of course from the higher level, but with the necessary (lower) limitations. (As such they are thought over on the higher level.) From this point of view, we understand the various meanings (levels) of the same words. Then, any mutual mixing and changing the metalanguage and language level or the auto‐reference (paradox, noetical paradox, contradiction and antinomian) is excluded.

2. Goedel numbers, information and thermodynamics

Any inference within the system P3 sets the TPAtheoretical relation4 among its formulae a[]. This relation is given by their gradually generated special sequence a=[a1, , aq, , ap, , ak, ak+1] which is the proof of the latest inferred formula ak+1. By this, the unique arithmetic relation between their Goedel numbers, FORMULAE x[], x[]=Φ(a[]), is set up, too. The gradually arising SEQUENCE of FORMULAE x=Φ(a) is the PROOF of its latest FORMULA xk+1.

Let us assume that the given sequence a=[ao1, ao2, , ao, , aq, , ap, , ak, ak+1] is 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.5

Within the process of the (goedelian) arithmetic‐syntactic analysis of the latest formula ak+1 of the proof a we use, from the a selected, (special) subsequence aq,p,k+1 of the formulae aq, ap, ak+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,

aq,p,k+1=[aq,ap,ak+1],  apaq  ak+1,  aq,p,k+1=[aq, aq  ak+1, ak+1],x=Φ(a)=Φ([Φ(a1), Φ(a2), , Φ(aq), , Φ(ap), , Φ(ak), Φ(ak+1)]])=Φ(x)=Φ(x1)Φ(x2)  Φ(xq)  Φ(xp)  Φ(xk)Φ(xk+1)l(x)=l[Φ(x)]=l[Φ(a)]=k+1,xk+1=Φ(ak+1)=l[Φ(a)]Gl Φ(a)=(k+1)Gl xxp=Φ(ap)=Φ(aqak+1)=qGl Φ(a)  Φ()  l[Φ(a)]Gl Φ(a)=qGl xImp [l(x)]Gl xxq=Φ(aq)=qGl Φ(a)=qGl x

Checking the syntactic and TPA‐theoretical correctness of the analyzed chains ai, as the formulae of the system P having been generated by inferring (Modus Ponens) within the system P (in the theory TPA), and also the special sequence of the formulae a of the system P (theory TPA), is realized by checking the 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()6; the core of the whole (goedelian) arithmetic‐syntactic analysis is the (procedure) relation of Divisibility,

Form[Φ(ai)]=1/0,  FR[Φ(a1i+1)]=1/0,  o  i  kOp[xk,Neg(xq), xk+1] = Op[Φ(ap),Φ[(aq)], Φ(ak+1)]=1/0Fl[(k+1)Gl x, pGl x, qGl x]=1/0xB xk+1=1/0,  Bew(xk+1)=1/0;Φ(ap)||233Gl Φ(aq,pk+1) & Φ(ap)||71Gl Φ(aq,pk+1)=1/0¯

2.1. Inference in the system P and information transfer

The syntactic analysis of the special sequence of the formulae a of the system P in general, and therefore, also its arithmetic‐syntactic version, that is the activity of (goedelian) arithmetic‐syntactic analyzer, will be expressed by means of terms of information transfer through a certain information transfer channel K.

As such, it is a sequence of successive attempts i to transfer information with input, loss and output messages [api,aqi,ai+1],[api,aqi] and [ai+1] with their information amounts J(aqi,pi,i+1),J(aqi,pi) and J(ai+1). Index i is a serial number of the inferencinganalyzingtransferring step, 0<qi<pi<i+1l[Φ(a)]=k+1. The Goedel numbering also enables us to consider the individual Goedel numbers xi, xi|yi and yi of messages [api,aqi,ai+1], [api,aqi] a [ai+1] as messages too, with their (and the same) information amounts J(xi), J(xi|yi) a J(yi),

[api,aqi,ai+1]aqi,pi,i+1xi=Φ(aqi,pi,i+1),  [api,aqi]aqi,pixi|yi=Φ(aqi,pi)[ai+1]ai+1yi=Φ(ai+1)Φ(aqi,pi,i+1)=Φ(aqi)Φ(api)Φ(ai+1)=Φ(aqi,pi)Φ(ai+1),  Φ(aqi,pi)=Φ(aqi)Φ(api);J(xi)=J[Φ(aqi,pi,i+1)], J(xi|yi)=J[Φ(aqi,pi)], J(yi)=J[Φ(ai+1)]    

For each ith step of the goedelian syntactic analysis, we determine the values


We regard these values as average values H(X), H(X|Y) and H(Y) of information amounts of message sources X, X|Y and Y with selective spaces X, X×Y and Y, and with the uniform probability distribution,

X=Def[X, πX(xi)=const.], card X=2Φ(aqi,pi,i+1),  πX(xi)=12Φ(aqi,pi,i+1)Y=Def[Y, πY(yi)=const.],   card Y=5Φ(ai+1),  πY(yi)=15Φ(ai+1)j=1card X12Φ(aqi,pi,i+1)=2Φ(aqi,pi,i+1)2Φ(aqi,pi,i+1)=1,  j=1card Y15Φ(ai+1)=5Φ(ai+1)5Φ(ai+1)=1

It is obvious that we consider a direct information transfer [11] through the channel K without noise, disturbing yi|xi, which means with the zero noise (disturbing) information [J(yi|xi)=0][H(Y|X)=0], [yi|xiΦ (null)].

In each ith step of the activity of our information model K of the arithmetic‐syntactic analysis, it is valid that X:=xi=Φ(aqi,pi,i+1) and Y:=yi=Φ(ai+1)=xi+1, and the channel equation is applicable [11],

T(X;Y)=H(X)H(X|Y)=H(Y)H(Y|X)=T(Y;X)  T(X;Y)=J(xi)J(xi|yi) =J(yi)J(yi|xi)=T(Y;X)  now in the formT(X;Y)=H(X)H(X|Y)=H(Y),  T(X;Y)=J(xi)J(xi|yi)=J(yi)

The relation Φ(aqi,pi,i+1)B Φ(ai+1) (xiB yi) is evaluated by the relation of Divisibility and we identify its execution7 with the actual direct information transfer in the channel K. So, when our inference by Modus Ponens is done correctly, in each ith step, we have its information interpretation, in steps i,

[xiB yi]  [J(xi)J(xi|yi)>0]  [T(xi;yi)>0]  [T(X;Y)>0][Fl(yi,xpi,xqi)]  Fl[Φ(ai+1),Φ(aqi),Φ(api)]  [Φ(aqi,pi,i+1)B Φ(ai+1)][Φ(api)||233Gl xi & Φ(api)||7xi]  [Φ(api)||233Gl Φ(aqi,pi,i+1) & Φ(api)||71Gl Φ(aqi,pi,i+1)]

Let us assume that, when inferring by Modus Ponens, b, [(b)(c)]c, we make such an error that we write b, [(b)(c)]d, dc where, however, the chain d (by chance) can also be (in the form of) a formula of the language LP of the system P.8 For the considered NOT‐INFERRABILITY of yi [=d], being interpreted now from the point of information view, we put J(Φ(ai+1)) =Def 0, or better said, with regard of the properties of INFERENCE, we are forced to put Φ(ai+1)=Def0 within the framework of the theory TPA and then, informationally

H(Y)=T(X;Y)=Defln[5Φ(ai+1)]=0,H(X)=H(X|Y) J(xi)J(xi|yi)=J(yi)=0, J(xi)=J(xi|yi)ηi =Def J(yi)J(xi)=H(Y)H(X),0ηi1

2.2. Thermodynamic consideration

The thermodynamic consideration of an information transfer [11] reveals that the input message aqi,pi,i+1 carries the input heat energy ΔQWi transformed by the reversible direct Carnot Cycle (Machine) C into the output mechanical work ΔAi corresponding to the output message ai+1. The heater A of the Carnot Cycle (Machine) C has the temperature TW and models the source of input messages (the message aqi,pi,i+1) of the channel K. Its cooler B has the temperature T0 determining the transfer efficiency ηi. By the value ηi>0 the fact of inferrability of the chain ai+1 from the special sequence of formulae aqi,pi,i+1 as the formula of the theory TPA is stated.

Thus, the reversible direct Carnot Cycle C is the thermodynamic model of the direct information transfer through the channel K [11], and hereby of the inferring (INFERRING) itself, and also of the arithmetic‐syntactic analysis of formulae of the language LTPA of the theory TPA.8 Thus, we have


Now we obtain the information formulation [11] of the changes of the heat (thermodynamic) entropies ΔSC[i], ΔSAB[i] and ΔSA[i] in the thermodynamic model C of our information transfer—inferring (INFERRING)—arithmetic‐syntactic analysis within the (language of the) system P,


In accordance with Ref. [11], it is valid that, within the inferring—arithmetic‐syntactic analysis—information transfer, the thermodynamic entropy SC of an isolated system, in which the modeling reversible direct Carnot Cycle C is running parallelly, increases in every ith step by the value ΔSC[i],

ΔSC[i]=kJ(ai+1)=kH(Y)),  H(Y)J(ai+1)=ΔA[i]kTW  0

Provided that the ith inferring step has been done and written correctly (Modus Ponens) the Goedel arithmetic‐syntactic analyzer decides, correctly, for the obtained a1i+1[a1i,ai+1], that the relations Φ(aqi,pi,i+1)B Φ(ai+1) [Φ(a1i+1)B Φ(ai+1)] and Bew[Φ(ai+1)] are valid, and the information‐thermodynamic model (KC) generates the non‐zero, positive output value T(X;Y) for the inferring step i [for X:=xi=Φ(aqi,pi,i+1) or X:=xi=Φ(a1i), respectively, and for Y:=yi=Φ(ai+1)],


The zero change of the whole heat entropy SC of the isolated system in which our model cycle C is running occurs just when in the inferential system P, from the perspective of the theory TPA, nothing is being inferred in the step i, ΔSC[i]=0. Now, particularly in that sense that we mistakenly apply the conclusion of the rule Modus Ponens and we declare it to be an inferring step. Then, from the point of view of the TPA‐inference, we do not exert any ‘useful effort’ or energy in order to derive a new TPA‐relation [ formula ai+1, FORMULA Φ(ai+1)]. The previous ‘effort’ or energy associated with our inference (no matter that TPA‐correct) of the sequence of aii is worthless. The formula ai+1 [=d] is just arbitrarily added to the previous sequence a1i of formulae of the theory TPA in such a way that it does not include any such formulae aqi and api that it would be valid Φ(api,qi,i+1)B Φ(ai+1)= 1. In the information‐thermodynamic interpretation, we write (for X:=xi, Y:=yi=d)

J(yi)=H(Y)=0J(xi)=H(X)=H(X|Y)=J(xi|yi)ηi=0 ΔSC[i]=0TW=T0ΔQWi=ΔQ0i   ηiΔQWi=kJ(yi)=0  ηi=0

We have not exerted any inferring energy within the framework of building up the theory TPA, in order to create information J(yi)>0, and then we necessarily have ηi=0, J(yi)=0 where ηi=0 expresses this error. All before ai+1, otherwise inferred correctly, is not related to it–the information transfer channel K is interrupted. The overall amount of our inference efforts exerted in vain up to ai included can be evaluated by the whole heat energy10

kH(X|Y)=kln[Φ(a1i)]=ln[2Φ(a1)3Φ(a2)  πiΦ(ai)]

3. Goedel substitution function and FORMULA 17Gen r

Let us consider the instance of the relation Q(X,Y) for the specific values x and y, X:=x and Y:=y, which is the constant relation Q(x,y), and let us define the Goedel numbers y and y that the Goedel (variable) number (his ‘CLASS’ SIGN) y arises from Admissible Substitution from the FORMULA q(17,19) [ the ARITHMETIZATION of Q(X,Y)],

y=Sb(17q(17,19)Z(x))=y(19) [=Φ[Q(x,Y)] ]andy=Sb(19yZ(y))

Any of the following notations can be used

q(u1,u2)=q(17,19)=Φ[Q(X,Y)]=Φ[q(u,v)]=Φ[Q(X,Y)]q(u1,u2)=q(17,19)=Φ[Q(X,Y)]q(u1,u2), q(17,19), q(u,v)Q(X,Y), q[Z(x),u2]=y(u2)=q[Z(x),19]=y(19)=y=Φ[Q(x,Y)]Q(x,Y)

The following Admissible Substitution Sb(19yZ(y)) is carried out in the second step of the given Double Substitution Sb(1719qZ(x)Z(y)); in the Goedel variable number q(17,19), we first put 17:=Z(x) and in the result q[Z(x),19] we put 19:=Z(y). Then


The CLAIM y only seems to be a constant P/TPAFORMULA, which, as the CLAIM y[Z(y)] speaks only about a common number y. But, by the NUMERAL Z(y)] it is the y speaking about y and then, it is the FORMULA y speaking about itself.

Let us think of the goedelian arithmetic‐syntactic generator, the job of which is to ‘print’ the Goedel numbers of the constant FORMULAE obtained by Admissible Substitutions of NUMERALS into their FREE VARIABLES (now of the Type1). In case of the ‘global’ validity of the substitution 19:=Z(y)11 it creates from the given FORMULA y an infinite sequence of semantically identical FORMULAE y [=y[Z(y)]], y[Z(y)] [=y[Z[y[Z(y)]]]], with the aim to end the process by ‘printing’ just the value y. But it never reveals this outcome y; however, we—metatheoretically—know it. It never gets as far as to print the natural number y which it ‘wants to reach’ by creating the infinite sequence of outcomes of the permanently repeated substitution 19:=Z(y) which prevents it from this goal (y marks the claim y about the claim y, the claim y about the claim y about the claim y etc.). It is even the first one, by which the analyzer is trying to calculate and ‘print’ y, that prevents it from this aim. We never obtain a constant Goedel number. The FORMULA y[Z(y)] arises by applying the (Cantor) diagonal argument, which is not any inference rule of the theory TPA (and of the system P), and thus, it is not an element of the language LTAP (and LP). This is the reason for not‐recursivity of the relations Bew(); the upper limit of its computing process is missing. First, we have q[Z(x),19]19:=Z(y)q[Z(x),Z(y)]=y[Z(y)]=y  and then 'try12

yq[Z(x),Z[q[Z(x),19]]]19:=Z(y)q[Z(x),Z[q[Z(x),Z(y)]]]=y[Z[y[Z(y)]]]q[Z(x),Z[q[Z(x),Z[q[Z(x),19]]]]]19:=Z(y)q[Z(x),Z[q[Z(x),Z[q[Z(x),Z(y)]]]]]q[Z(x),Z[q[Z(x),Z[q[Z(x),Z[q[Z(x),19]]]]]]]19:=Z(y)q[Z(x),Z[q[Z(x),Z[q[Z(x),Z[q[Z(x),Z(y)]]]]]]]q[Z(x),Z[q[Z(x),Z[q[Z(x),Z[q[Z(x),Z[q[Z(x),19]]]]]]]]]19:=Z(y)    ad lib.

It is obvious that the Substitution function, no matter how much its execution complies with the recursive grammar, is not total and, therefore, nor recursive. For this reason, it is convenient to redefine it as a total function and, therefore, also recursive one and to put [y[Z(y)]]=0 but, due to the inference properties, Neg[y[Z(y)]]=0 too. Then

Sb(19y(19)Z(y))=Def0¯¯&Sb(19Neg[y(19)]Z(y))=Sb(19y(19)Z(y))¯=Def0¯¯Bew[y[Z(y)]]=Bew(0)=0¯¯&Bew[Neg[y[Z(y)]]]=Bew(0)=0¯¯Q(x,y)xB[Sb(19yZ(y))]¯=q[Z(x),Z(y)]=y[Z(y)]=yxB y¯

Also see the Proposition V in Refs. [3, 4, 5]. The mere grammar derivation, writability convenient to the recursive grammar is quite different from the TPA‐provability. The Goedel number y, the FORMULA y[Z(y)], is seemingly a FORMULA (and even constant) of the system P and thus it is not an element of the theory TPA; is not of an arithmetic type (it is not recursive arithmetically, only as for its basic syntax, syntactically). As the CLAIM y[Z(y)] it speaks about the number y only, but by that it is the number y itself, then as y[Z(y)], it claims its own property, that from the Goedel number x it itself IS NOT INFERRED within the system P [Bew(y)=0]. It is true for the given x and it ‘says’: ‘I, FORMULA y[Z(y)], am in the system P (by it means) from the Goedel number x UNPROVABLE.’ And, by this, it also states both the property of the system P and the theory TPA.

3.1. FORMULA 17Gen r and information transfer

With regard of the fact that FORMULA y is constructed by the diagonal argument, it is not INFERRED within the system P—in the TPA and so, it is not provable for any x from 0. Then, within the framework of the theory TPA, we put 17Gen y=Def0¯¯ and thus J(17Gen y)=Def0¯¯.13 In the proof we put p:17Gen q, [17u1X,19u2Y, q=q(17,19)], and then, in compliance with the Goedel notation,

p=17Gen q(17,19)=Φ[u1Πq(u1,u2)][=Φ[xX|Q(x,Y)]Q(X,Y)]Q(0,Y)

The metalanguage symbol Q(X,Y) in (20) or the symbol Q(0,Y) is read as follows:

‘None xX(0) is in the relation INFERENCE to the content (to the selective space Y) of the variable Y. From any given x,x=Φ(a)=Φ([a1k,ak+1]), xX(0), any Goedel number Φ(ak+1)0, writable as the proposed outcome of the INFERENCE from the given x, is NOT INFERRED in reality.’

We put r:=Sb(19qZ(p))=Sb(19q(17,19)Z(p))=Sb(19q(17,19)Z[17Gen q(17,19)]).

The Goedel number r, rr(17)=Φ[Q(X,p)] is, by the substitution Z(p), supposingly [3, 4, 5], the CLASS SIGN with the FREE VARIABLE 17, but also remains be the variable Goedel number in the VARIABLE 19. It contains the FREE VARIABLE 19 as hidden and 17 is both FREE and BOUND in it, [q[17,Z[17Gen q(17,19)]],

r=r(17)=q[17,Z[p(19)]]=q[17,Z[17Gen q(17,19)]]q[u1,Z(p)]Q(X,p) =q[u1,Φ[u1Πq(u1,u2)]]=Φ[Q[X,Φ[xX|Q(x,Y)]]],  Q(X,p)=Q(X,Y)Y:=pQ[X,Φ[Q(X,Y)]]Q[X,Φ[Q(0,Y)]]

Further14 Q(X,Y)X:=x=Q(x,Y),Q(x,Y)Y:=p=Q(x,p) and then,

r[Z(x)]=r(17)17:=Z(x)q[17,Z(p)]17:=Z(x)=q[Z(x), Z[17Gen q(17,19)]]= q[Z(x), Z(p)]] = q[Z(p)]= q=q[Z(x),Z[Φ[u1Πq(u1,u2)]]]=Φ[Q[x,Φ[xX|Q(x,Y)]]]=Φ[Q[x,Φ[Q(X,Y)]]]=Φ[Q[x,Φ[Q(0,Y)]]]

With regard of quantification r[Z(x)] over values Z(x) of the variable u1, we write

Z(x)Gen r[Z(x)]=Z(x)Gen q[Z(x),Z(p)]]=Z(x)Gen p[Z(p)]=Z(x)Gen q=Z(x)Gen q[Z(x),Z[17Gen q(17,19)]]=p[Z(p)]=p17Gen q[17,Z[17Gen q(17,19)]]=17Gen r(17)=17Gen q[17,Z[p(19)]]=17Gen q[17,Z[17Gen q(17,19)]]=Φ[u1Π[Φ[q[u1,Φ[u1Πq(u1,u2)]]]]]=Φ[xX|Φ[Q[x,Φ[xX|Q(x,Y)]]]]=17Gen rQ(X,p)=Q(X,Y)Y:=pQ[X,Φ[Q(X,Y)]]=Q[0,Φ[Q(0,Y)]]

The relation Q(X,p), Q(X,p)=xX|Q[x,Φ[xXQ(x,p)]] and, therefore, the relation T(X,p)¯ says that no such x exists to comply with the message transfer conditions of p from x; the infinite cycle is stipulated. Attempts to give the proof of the FORMULA 17Gen r within the framework of the inferential system P, that is, attempts to ‘decide’ it inside the system P only by the means of the system P itself end up in the infinite cycle.

The claim 17Gen r does not belong to the theory TPA but gives a witness about it—about its property. It is so because it is formulated in a wider/general formulative language LP* than the language LP of the system P and so outside both of the language LP (and as such, outside of the language LTAP too). The FORMULAE/CLAIMS of both the theory TPA and the system P speak only about finite sets of arithmetic individuals but the theory TPA and the system P are the countable–N0‐sets.15 It seems only that 17Gen r is a part (of the ARITHMETIZATION) of the theory TPA and of the system P which is by it is written down (grammatically only) according to the common/general recursive syntax of the general formulative language LP* in which all the arithmetic relations are written (and, in addition, the TPA‐relations are inferred). On the other hand, there nothing special on its evaluation, but from the point of view or position of the metalanguage only (!). From the formalistic point of view, it is a number only. From the semantic point of view, it is an arithmetic code but of the not‐arithmetic claim.16

Let the Goedel number t[Z(x),Z(y)] be DESCRIPTION of the mechanism of the transfer y from x (on the level of the system P and the theory TPA) in the channel K,

Sb(1719tZ(x)Z(y))Subst tK(U1,U2)[U1U2J(x)J(y)][J(x)J(y)J(x)]

But, when it is valid that Sb(19yZ(y))=0¯¯=Sb(1719qZ(x)Z(y)) then the number y is not a FORMULA of the system P and in the information interpretation of inferring (INFERRING) within the system P it is valid that, J(y)=0¯¯. Then we can consider the simultaneous validity of [J(y)>0]&[J(y)<0]¯¯—also see the Proposition V in Refs. [3, 4, 5], which, from the thermodynamic point of view, means the equilibrium and, from the point of computing, the infinite cycle [14, 16]. For the information variant of the FORMULA 17Gen r and Goedel number p=p[Z(p)] is valid

p=DefSubst p(U2)(U2J(p))=Subst U1Gen qK(U1,U2)(U2J[U1Gen qK(U1,U2)])p=17Gen qK[U1,J[U1Gen qK(U1,U2)]]=U1Gen qK[U1,J(p)]=p[J(p)] U1Gen r(U1) = U1Gen r u1Π[qK[u1,J[u1ΠqK(u1,u2)]]]]=xX|QK[X,Φ[xX|QK(x,Y)]] =QK(X,p)]=QK(X,Y)Y:=p=QK[X,Φ[QK(X,Y)]]=QK[0,Φ[QK(0,Y)]]

So, the message p (the message p about itself) is not‐transferrable from any message x,

[xB[K] p¯= 1][xB[K] p¯= 1][τ[K](x,y)= 0][J(p)=0][J(p)=0]

It is the attempt to transfer the message y (y=17Gen r) through the channel K, while this message itself causes its interruption and ‘wants’ to be transferred through this interrupted channel K as well.17 Its ‘errorness’ is in our awaiting of the non‐zero outcome J(y)>0 when it is applied in the (direct) transfer scheme K because the information J(y)>0, y=17Gen r (known from and valid in the metalanguage), from the point of transferrability through the channel K (from the point of inferrability in the theory TPA) does not exist. In the theory, TPA is J(y)=0 for the CLAIM 17Gen r is not arithmetic at all, it is the metaarithmetic one. From the point of the theory TPA and the system P, it is not quite well to call CLAIM 17Gen r as the SENTENTIAL FORMULA; it has only such form. For this reason, we use the term CLAIM 17Gen r or ‘SENTENTIAL FORMULA’/‘PROPOSITION.’

The message about that the channel K is for y interrupted cannot be transferred through the same channel K interrupted for y (however, through another one, uninterrupted for y, it can). Or we can say that the claim ak+1 [CLAIM y, y=Φ(ak+1)=17Gen r] is not inferable (INFERABLE) in the given inferential system P (but in another one making its construction‐INFERENCE possible, it is),


By constructing the FORMULA 17Gen r and from the point of information transfer, we have produced the claim ‘the transfer channel K is from p and on interrupted.’ Or, we have made the interrupted transfer channel directly by this p when we assumed it belonged to the set of messages transferrable from the source X. So, first we interrupt the channel K for p, and then, we want to transfer this p from the input x which includes this p (or is identical to it), and so the internal and input state of the channel K are (also from the point of the theory TPA) equivalent informationally. It is valid that J(p)=0¯¯ for any x, xX [ so cX|[J(p)=0]],

xX|[J(x)=J(x|p)][J(X|p)=J(X)>0] and for the simplicity is J(p|X)=0[xX|τ(x,p)¯][xX|[J(x)J(x|p)>0]¯][x0|[J(x)J(x|p)>0]¯]= 1x=Φ(a1k)17Gen r=Φ(a1k)Sb(19pZ(p))=Φ([a1k],p)Φ([a1k],null)J(x)=J[Φ(a1k)Φ(0)]=J[Φ([a1k])·20]=J(x|p)=J[Φ([a1k])],  x|p=Φ([a1k])

The channel K, however, always works only with the not zero and the positive difference of information amounts J(x)J(x|y) and in the theory TPA now it is valid that J(y)=J(x)J(x|p)=J(null)=0, J(y)=J(p)=J(null)=018. It means that our assumption about p [=r] is erroneous. No input message x having a relation to the output message p exists. The FORMULA 17Gen r both creates and describes behavior of the not functioning (interrupted) information transfer, from p on further. For the efficiency η of the information transfer, it is then valid [14, 16] that

The CLAIM (‘SENTENTIAL PROPOSITION’) 17Gen r we interpret as follows:

  • No information transfer channel K transfers its (internal) state x|y [ the information J(x|y)] given as its input message x, it behaves as interrupted.

  • There is no x0 for which it is possible to generate the Goedel number Φ[Q(0,Y)] which claims that there is no x0 for which it is possible to generate the non‐zero Goedel number y that we could write into the variable Y. This means that from any Goedel number x no INFERENCE is possible just for its latest part y=Φ(ak+1)=17Gen r has not been INFERRED either.

The metaarithmetic sense of the CLAIM (‘SENTENTIAL PROPOSITION’) 17Gen r is:

  • Within the general formulative language LP* of the inconsistent metasystem P (containing the consistent subsystem P with the theory the TPA) it is possible to construct [ be the (Cantor) diagonal argument ] such a claim (with the Goedel arithmetization code 17Gen r) which is true, but both this claim i and its negation are not provable/PROVABLE by the means of the system P (in the system P) and thus, also in the theory TPA–they are the meta‐TPA and the meta‐P claims not belonging to the system P, but they belong to the inconsistent system P, to its part PP (PP).

  • So, the Goedel Proposition VI (1931) [3, 4, 5] should be, correctly, ‘For the system exists ’ (which Goedel also, but not uniquely says), ‘For the theory exists, (nevertheless outside of them); by the author’s conviction the error is to say.’ In each consistent (?) system exists or, even ‘In the consistent (?) theory exists .’

4. 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 Goedel formula and its negation are not provable and decidable within it. They are constructed, not inferred, by the (Cantor) 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 Goedel 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 Goedel 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 Goedel 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 Goedel 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 (of the information transfer).

We have shown that the CLAIM/‘FORMULA’ 17Gen r, no matter how much it complies with the grammar of recursive writing of TPA‐arithmetic FORMULAE, is not such a FORMULA; it is not an element of the theory TPA and in convenience with [1, 2, 6, 7, 18, 19] nor an element of the system P19 and neither is r. The same is for Neg(17Gen r) (it cannot be inferred in P for is not inferable in P.) Nevertheless, we are in accordance with the intuitive and obviously intended sense of the Goedel Proposition VI20 which we, as the metalanguage one, have proved by metalanguage (information‐thermodynamic‐computing) means. We see, with our correction, that the CLAIMS (the Goedel ‘SENTENTIAL PROPOSITIONS’/‘FORMULAE’) 17Gen r, Neg(17Gen r) and the Proposition VI as the claim about them are metaarithmetic (methodological) statements.

5. Appendix

5.1. Auto-reference in information transfer, self-observation

In any information transfer channel K the channel equation


it is valid [?]. This equation describes the mutual relations among information entropies [(average) information amounts] in the channel K.

The quantities H(X), H(Y), H(X|Y) and H(Y|X) are the input, the output, the loss and the noise entropy.

The difference H(X)H(X|Y) or the difference H(Y)H(Y|X) defines the transinformation T(X;Y) or the transinformation T(Y;X), respectively,

H(X)H(X|Y)T(X;Y)   =   T(Y;X)H(Y)H(Y|X)

When the channel K transfers the information (entropy) H(X), but now just at the value of the entropy H(X|Y), H(X)=H(X|Y), then, necessarily, must be valid

T(X;Y)=0   [=H(Y)H(Y|X)]
  • For H(Y|X)=0, we have T(X;Y)=H(Y)=0.

  • For H(Y|X)0 we have H(Y)=H(Y|X)0

In both these two cases, the channel K operates as the interrupted (with the absolute noise) and the output H(Y) is without any relation to the input H(X) and, also, it does not relate to the structure of K. This structure is expressed by the value of the quantity H(X|Y). We assume, for simplicity, that H(Y|X)=0.

From Eqs. (30) to (32) follows that the channel K cannot transfer (within the same step p of its transfer process) such an information which describes its inner structure and, thus, it cannot transfer—observe (copy, measure) itself. It is valid both for the concrete information value and for the average information value, as well.

Any channel K cannot transfer its own states considered as the input messages (within the same steps p). Such an attempt is the information analogy for the Auto‐Reference known from Logics and Computing Theory. Thus, a certain ‘step‐aside’ leading to a non‐zero transfer output, H(Y)=H(X)H(X|Y)>0, is needed. (For more information see [14, 15, 16].

5.2. Auto-reference and thermodynamic stationarity

The transfer process running in an information transfer channel K is possible to be comprehended (modeled or, even, constructed) as the direct Carnot Cycle O [8, 10]. The relation OK is postulated. Further, we can imagine its observing method, equivalent to its ‘mirror’ OK. This mirror O is, at this case, the direct Carnot Cycle O as for its structure, but functioning in the indirect, reverse mode [8, 10].

Let us connect them together to a combined heat cycle OO in such a way that the mirror (the reverse cycle O) is gaining the message about the structure of the direct cycle O. This message is (carrying) the information H(X|Y) about the structure of the transformation (transfer) process (OK) being ‘observed.’ The mirror OK is gaining this information H(X|Y) on its noise ‘input’ H(Y|X) [while H(X)=H(Y) is its input entropy].

The quantities ΔQW, ΔA and ΔQ0 or the quantities ΔQW, ΔA and ΔQ0, respectively, define the information entropies of the information transfer realized (thermodynamically) by the direct Carnot Cycle O or by the reverse Carnot Cycle O (the mirror), respectively, (the combined cycle OO is created),

H(X)=ΔQWkTW,  resp.   H(Y)=ΔQWkTW H(Y)=ΔAkTW,  resp.  H(X)=ΔAkTWH(X|Y)=ΔQ0kTW,  resp.   H(Y|X)=ΔQ0kTW

Our aim is to gain the non‐zero output mechanical work ΔA* of the combined heat cycle OO, ΔA*>0. We want to gain non‐zero information H*(Y*)=ΔA*kTW>0.

To achieve this aim, for the efficiencies ηmax and η max of the both connected cycles O and O (with the working temperatures TW=TW and T0=T0, TWT0>0), it must be valid that ηmax>ηmax; we want the validity of the relation21


When ΔQ0=ΔQ0 should be valid, then must be that ΔQW<ΔQW  [(ηmax>ηmax)], and thus, it should be valid that

 ΔA*=ΔQWηmaxΔQWηmax>0 butΔQWηmaxΔQWηmax=ΔQ0ΔQ0=0

Thus, the output work ΔA*>0 should be generated without any lost heat and by the direct change of the whole heat ΔQWΔQW but within the cycle OO. For ηmax<ηmax the same heat ΔQWΔQW should be pumped from the cooler with the temperature T0 to the heater with the temperature TW directly, without any compensation by a mechanical work. We see that ΔA*=0 is the reality.

Our combined machine OO should be the II. Perpetuum Mobile in both two cases. Thus, ηmax=ηmax must be valid (the heater with the temperature TW and the cooler with the temperature T0 are common) that

ηmax=ηmax<1  and then  ΔQW=ΔQW

We must be aware that for ηmax=ηmax<1 the whole information entropy of the environment in which our (reversible) combined cycle OO is running changes on one hand by the value


and on the other hand it is also changed by the value H(X)ηmax=ΔQWkTW(1β) Thus, it must be changed by the zero value


The whole combined machine or the thermodynamic system with the cycle OO is, when the cycle OO is seen, as a whole, in the thermodynamic equilibrium. (It can be seen as an unit, analogous to an interruptable operation in computing.)

Thus, the observation of the observed process O by the observing reverse process O with the same structure (by itself), or the Self‐Observation, is impossible in a physical sense, and, consequently, in a logical sense, too (see the Auto‐Reference in computing).

Nevertheless, the construction of the Auto‐Reference is describable and, as such, is recognizable, decidable just as a construction sui generis. It leads, necessarily, to the requirement of the II. Perpetuum Mobile functionality when the requirements (34) and (35) are sustained.

(Note that the Carnot Machine itself is, by its definition, a construction of the infinite cycle of the states of its working medium and as such is identifiable and recognizable.) For the methodological step demonstrating the Information Thermodynamic Concept Removing see [14, 15, 16].

5.3. Gibbs paradox - auto-reference in observation

Only just by a (thought) ‘dividing’ of an equilibrium system A by diaphragms [9, 10, 11, 13], without any influence on its thermodynamic (macroscopic) properties, a non‐zero difference of its entropy, before and after its ‘dividing,’ is evidenced.

Let us consider a thermodynamic system A in volume V and with n matter units of ideal gas in the thermodynamic equilibrium. The state equation of A is pV=nRΘ. For an elementary change of the internal energy U of A, we have dU=ncvdΘ.

From the state equation of A, and from the general law of energy conservation [for a (substitute) reversible exchange of heat δq between the system and its environment], we formulate the I. Principle of Thermodynamics, δq=dU+pdV

From this principle, and from Clausius equation ΔS=DefΔqΘ, Δq=cvΔΘ+RΘΔVV, Θ>0, follows that


Let us ‘divide’ the equilibrial system A in a volume V and at a temperature Θ, or, better said, the whole volume V (or, its whole state space) occupiable, and just occupied now by all its constituents (particles, matter units), with diaphragms (thin infinitely, or, ‘thought’ only), not affecting thermodynamic properties of A supposingly, to m parts Ai, i{1, …, m}, m1 with volumes Vi with matter units ni. Evidently n=i=1mni and V=i=1mVi.

Let now S0(n)=0 and S0i(ni)=0 for all i. For the entropies Si of Ai considered individually, and for the change ΔS, when volumes V, Vi are expressed from the state equations, and for p=pi, Θ=Θi it will be gained that σ[i]=Rn[i]lnn[i]. Then, for Si=σi=ni(cvlnΘ+RlnVi) is valid, we have that

i=1mSi=i=1mσi=ncvlnΘ+Rln(i=1mVini), ΔS=Si=1mSi=σi=1mσi=Δσ=RlnVni=1mVini=nRi=1mninlnnin>0

Let us denote the last sum as B further on, B<0. The quantity B expressed in (40) is information entropy of a source of messages with an alphabet [n1, n2,  ,nm] and probability distribution [nin]i=1m. Such a division of the system to m parts defines an information source with the information entropy with its maximum lnm.

The result (37), ΔS=nRB, is a paradox, a contradiction with our presumption of not influencing a thermodynamic state of A by diaphragms, and, leads to that result that the heat entropy S (of a system in equilibrium) is not an extensive quantity. But, by the definition of the differential dS, this is not true.

Due to this contradiction, we must consider a non‐zero integrating constants S0(n), S0i(ni), in such a way, that the equation ΔS=(σ+S0)i=1m(σi+S0i)=0 is solvable for the system A and all its parts Ai by solutions S0[i](n[i])=n[i]Rlnn[i]γ[i].

Then, S[i]S[i]Claus, and we write and derive that


Now let us observe an equilibrium, S=SClaus=SBoltz=kNB*=kNlnN.

Let, in compliance with the solution of Gibbs Paradox, the integration constant S0 be the (change of) entropy ΔS which is added to the entropy σ to figure out the measured entropy SClaus of the equilibrium state of the system A (the final state of Gay‐Lussac experiment) at a temperature Θ. We have shown that without such correction, the less entropy σ is evidenced, σ=SClausΔS,  ΔS=S0.

Following the previous definitions and results, we have

ΔS=ΔQ0Θ=nRlnnγ,lnγ=ΔSknNA+lnn=ΔSkN+lnNlnNA,  γ=N    ΔSkN=lnNA.

By the entropy ΔS the ‘lost’ heat ΔQ0 (at the temperature Θ) is defined.

Thus, our observation can be understood as an information transfer T in an information channel K with entropies H(X), H(Y), H(X|Y) and H(Y|X) in (33) but now bound physically; we have these information entropies per one particle of the observed system A:

 input   H(X)=DefSkN=lnγ=B*=lnN=rB(r)  output  H(Y)=DefσkNBGibbs=BBoltz=B(r),  loss  H(X|Y)=DefS0kN, noise H(Y|X)=Def0  for the simplicity ;H(X|Y)=rB(r)[B(r)]=B(r)(r1)=(B*)r1r, r1;  1r=ηmax.

For a number m of cells of our railings in the volume V with A, mN or for the accuracy r of this description of the ‘inner structure’ of A (a thought structure of V with A) and for the number q of diaphragms creating our railings of cells and constructed in such a way that q <1, m1>, we have that r=N1q.

Our observation of the equilibrium system A, including the mathematical correction for Gibbs Paradox, is then describable by the Shannon transfer scheme [X, K, Y], where

H(X)=SClauskN,  H(X|Y)=S0kN,  H(Y)=SClauskN,  H(Y|X)=ΔSkN.

However, a real observation process described in (44), equivalent to that one with r=1, is impossible.

We conclude by that, the diminishing of the measured entropy value about ΔS against S awaited, evidenced by Gibbs Paradox, does not originate in a watched system itself. Understood this way, it is a contradiction of a gnozeologic character based on not respecting real properties of any observation [8, 9, 10].

With our sustaining on the ‘fact’ of the Gibbs Paradox reality also mean the circulating value of ΔS (in our brain) just depending on our starting point of thinking about the observed system with or without the (thought) railings. Simultaneously (&) and in the cycle our brain would have [ΔS<0]& [ΔS>0]–see the validity of the Goedel Proposition V [3, 4, 5] for the inconsistent system P.

This and, also, Figure 1, is the thermodynamic equivalent to the paradoxical understanding to the Goedel Incompleteness Theorems, also known as the Goedel Paradox. In fact, both paradoxes do not exist in the described reality—they are in our brain, caused by the mixing of (our) consideration levels (the higher or methodology level and the lower or object/theoretical level) and, also, reveal themselves as the contradictions (on the lower level).


Figure 1.

Stationarity of the double cycle OO.


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


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1 The reader of the paper should be familiar with the Goedel proof’s way and terminology; SMALL CAPITALS in the whole text mean the Goedel numbers and working with them. This chapter is based, mainly, on Ref. [17], which was improved as for certain misprints, and also, by a few more adequate formulations and by adding the part Appendix [14, 15, 16].

2 B. Russel, L. Whitehead, Principia Mathematica, 1910, 1912, 1913 and 1927.

3 Formal arithmetic inferential system.

4 Peano Arithmetics Theory.

5 For simplicity. The ‘real’ inference is applied to the formula ai+1 for i=o.

6 Formula, Reihe von Formeln, Operation, Folge, Glied, Beweis, Beweis, see Definition 1–46 in Refs. [3, 4, 5] and by means of all other, by them ‘called’, relations and functions (by their procedures).

7 And of the other relevant procedures too, see definitions 1–46 in Refs. [3, 4, 5].

8 We just think mistakenly that dai+1 but ai+1=c is correct. Then the relation of Divisibility is not met. Neither is the relation of the Immediate Consequence.

9 Formulated in the language LP of the system P in compliance with its (with the TPA) inference rules.

10 πi is the i‐th prime number.

11 Caused by the application of the (Cantor) diagonal argument.

12 By substitution 19:=Z(y) nothing changes in variability of FORMULA y by the VARIABLE 19. The number y should denote infinite and not recursive subset of natural numbers or to be equal to them.


14 And similarly for Q(X,Y)Y:=p=Q(X,p), Q(X,p)X:=x=Q(x,p). It depends neither on the sequence of substitution steps nor on the sequence of operations Sb() and [] Gen [].

15 We have, inside of them, only N0 symbols for denoting their relations/formulae (or sets denoted by these relations/formulae). Thus, the CLAIM 17Gen r speaks about the element of the set with the cardinality N1 containing, as its elements, the N0‐sets; thus it can speak about the theory TPA, N0<N1 and cannot be in it or in the system P.

16 Thus it is not a common number as the [3, 4, 5] claims and neither is r.

17 In fact, it represents the very core of the sense of the Halting Problem task in the Computational Theory.

18 Attention (!) but x contains the message p that J(x)=J(x|p).

19 In the contrast to Refs. [3, 4, 5].

20 Because, on the other hand, Goedel 1931 [3, 4, 5] also says, correctly, ‘For the system exists …,’ ‘For the theory exists …, (nevertheless outside of them ‐ the author’s remark); the error is to say in the system exists …, in the theory exists ….’

21 We follow the proof of physical and thus logical impossibility of the construction and functionality of the Perpetuum Mobile of the II. and, equivalently [10], of the I. type.