Open access peer-reviewed chapter

Information Transfer and Thermodynamic Point of View on Goedel Proof

By Bohdan Hejna

Submitted: November 14th 2016Reviewed: March 27th 2017Published: December 20th 2017

DOI: 10.5772/intechopen.68809

Downloaded: 460


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.


  • 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,FORMULAEx[], 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+1of the proof awe 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,

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 xE1

Checking the syntactic and TPA‐theoretical correctness of 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 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¯E2

2.1. Inference in the system Pand information transfer

The syntactic analysis of the special sequence of the formulae aof the system Pin 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 ito 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 iis 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|yiand yiof 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)]    E3

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|Yand Ywith selective spaces X, X×Yand 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)=1E5

It is obvious that we consider a direct information transfer [11] through the channel Kwithout 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 Kof 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)E6

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)]E7

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, dcwhere, however, the chain d(by chance) can also be (in the form of) a formula of the language LPof 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)=Def0within the framework of the theory TPAand 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),0ηi1E8

2.2. Thermodynamic consideration

The thermodynamic consideration of an information transfer [11] reveals that the input message aqi,pi,i+1carries the input heat energy ΔQWitransformed by the reversible direct Carnot Cycle (Machine) Cinto the output mechanical work ΔAicorresponding to the output message ai+1. The heater Aof the Carnot Cycle (Machine) Chas the temperature TWand models the source of input messages (the message aqi,pi,i+1) of the channel K. Its cooler Bhas the temperature T0determining the transfer efficiency ηi. By the value ηi>0the fact of inferrability of the chain ai+1from the special sequence of formulae aqi,pi,i+1as the formula of the theory TPAis stated.

Thus, the reversible direct Carnot Cycle Cis 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 LTPAof 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 Cof 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 SCof an isolated system, in which the modeling reversible direct Carnot Cycle Cis 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  0E11

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 SCof the isolated system in which our model cycle Cis 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 aiiis worthless. The formula ai+1 [=d]is just arbitrarily added to the previous sequence a1iof formulae of the theory TPAin such a way that it does not include any such formulae aqiand apithat 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=0E13

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)=0where ηi=0expresses this error. All before ai+1, otherwise inferred correctly, is not related to it–the information transfer channel Kis interrupted. The overall amount of our inference efforts exerted in vain up to aiincluded can be evaluated by the whole heat energy10

kH(X|Y)=kln[Φ(a1i)]=ln[2Φ(a1)3Φ(a2)  πiΦ(ai)]E14

3. Goedel substitution function andFORMULA17Gen r

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

y=Sb(17q(17,19)Z(x))=y(19) [=Φ[Q(x,Y)] ]andy=Sb(19yZ(y))E15

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)E16

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 yonly 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 yspeaking about yand then, it is the FORMULA yspeaking 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 yan 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 ywhich 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 (ymarks the claim yabout the claim y, the claim yabout the claim yabout the claim yetc.). 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.E18

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)]]=0but, due to the inference properties, Neg[y[Z(y)]]=0too. 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¯E19

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 Pand 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 yonly, but by that it is the number yitself, then as y[Z(y)], it claims its own property, that from the Goedel number xit itself IS NOT INFERRED within the system P[Bew(y)=0]. It is true for the given xand it ‘says’: ‘I, FORMULA y[Z(y)], am in the system P(by it means) from the Goedel number xUNPROVABLE.’ And, by this, it also states both the property of the system Pand the theory TPA.

3.1. FORMULA 17Gen rand information transfer

With regard of the fact that FORMULA yis constructed by the diagonal argument, it is not INFERRED within the system P—in the TPAand so, it is not provable for any xfrom 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)E20

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)]]E21

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)]]]E22

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)]]E23

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 xexists to comply with the message transfer conditions of pfrom x; the infinite cycle is stipulated. Attempts to give the proof of the FORMULA 17Gen rwithin the framework of the inferential system P, that is, attempts to ‘decide’ it inside the system Ponly by the means of the system Pitself end up in the infinite cycle.

The claim 17Gen rdoes not belong to the theory TPAbut 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 LPof the system Pand so outside both of the language LP(and as such, outside of the language LTAPtoo). The FORMULAE/CLAIMS of both the theory TPAand the system Pspeak only about finite sets of arithmetic individuals but the theory TPAand the system Pare the countable–N0‐sets.15 It seems only that 17Gen ris a part (of the ARITHMETIZATION) of the theory TPAand of the system Pwhich 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 yfrom x(on the level of the system Pand 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)]E24

But, when it is valid that Sb(19yZ(y))=0¯¯=Sb(1719qZ(x)Z(y))then the number yis not a FORMULA of the system Pand in the information interpretation of inferring (INFERRING) within the system Pit is valid that, J(y)=0¯¯. Then we can consider the simultaneous validity of [J(y)>0]&[J(y)<0]¯¯—also see the Proposition Vin 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 rand 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 ru1Π[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)]]E25

So, the message p(the message pabout 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]E26

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 Kas well.17 Its ‘errorness’ is in our awaiting of the non‐zero outcome J(y)>0when it is applied in the (direct) transfer scheme Kbecause 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)=0for the CLAIM 17Gen ris not arithmetic at all, it is the metaarithmetic one. From the point of the theory TPAand the system P, it is not quite well to call CLAIM 17Gen ras the SENTENTIAL FORMULA; it has only such form. For this reason, we use the term CLAIM 17Gen ror ‘SENTENTIAL FORMULA’/‘PROPOSITION.’

The message about that the channel Kis for yinterrupted cannot be transferred through the same channel Kinterrupted 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 rand from the point of information transfer, we have produced the claim ‘the transfer channel Kis from pand on interrupted.’ Or, we have made the interrupted transfer channel directly by this pwhen we assumed it belonged to the set of messages transferrable from the source X. So, first we interrupt the channel Kfor p, and then, we want to transfer this pfrom the input xwhich includes this p(or is identical to it), and so the internal and input state of the channel Kare (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])E28

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 TPAnow 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 xhaving a relation to the output message pexists. The FORMULA 17Gen rboth creates and describes behavior of the not functioning (interrupted) information transfer, from pon further. For the efficiency ηof the information transfer, it is then valid [14, 16] that


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

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

  • There is no x0for which it is possible to generate the Goedel number Φ[Q(0,Y)]which claims that there is no x0for which it is possible to generate the non‐zero Goedel number ythat we could write into the variable Y. This means that from any Goedel number xno INFERENCE is possible just for its latest part y=Φ(ak+1)=17Gen rhas not been INFERRED either.

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

  • Within the general formulative language LP*of the inconsistent metasystem P(containing the consistent subsystem Pwith 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‐TPAand the meta‐Pclaims 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 TPAand 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 Pfor 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 VIas the claim about them are metaarithmetic (methodological) statements.

5. Appendix

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

In any information transfer channel Kthe 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)E31

When the channel Ktransfers 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)]E32
  • For H(Y|X)=0, we have T(X;Y)=H(Y)=0.

  • For H(Y|X)0we have H(Y)=H(Y|X)0

In both these two cases, the channel Koperates 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 Kcannot transfer (within the same step pof 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 Kcannot 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 Kis possible to be comprehended (modeled or, even, constructed) as the direct Carnot Cycle O[8, 10]. The relation OKis postulated. Further, we can imagine its observing method, equivalent to its ‘mirror’ OK. This mirror Ois, at this case, the direct Carnot Cycle Oas for its structure, but functioning in the indirect, reverse mode [8, 10].

Let us connect them together to a combined heat cycle OOin 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 OKis 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, ΔAand ΔQ0or the quantities ΔQW, ΔAand ΔQ0, respectively, define the information entropies of the information transfer realized (thermodynamically) by the direct Carnot Cycle Oor by the reverse Carnot Cycle O(the mirror), respectively, (the combined cycle OOis created),

H(X)=ΔQWkTW, resp.  H(Y)=ΔQWkTWH(Y)=ΔAkTW, resp. H(X)=ΔAkTWH(X|Y)=ΔQ0kTW, resp.  H(Y|X)=ΔQ0kTWE33

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 ηmaxand η maxof the both connected cycles Oand O(with the working temperatures TW=TWand T0=T0, TWT0>0), it must be valid that ηmax>ηmax; we want the validity of the relation21


When ΔQ0=ΔQ0should be valid, then must be that ΔQW<ΔQW  [(ηmax>ηmax)], and thus, it should be valid that

 ΔA*=ΔQWηmaxΔQWηmax>0 butΔQWηmaxΔQWηmax=ΔQ0ΔQ0=0E35

Thus, the output work ΔA*>0should be generated without any lost heat and by the direct change of the whole heat ΔQWΔQWbut within the cycle OO. For ηmax<ηmaxthe same heat ΔQWΔQWshould be pumped from the cooler with the temperature T0to the heater with the temperature TWdirectly, without any compensation by a mechanical work. We see that ΔA*=0is the reality.

Our combined machine OOshould be the II.Perpetuum Mobile in both two cases. Thus, ηmax=ηmaxmust be valid (the heater with the temperature TWand the cooler with the temperature T0are common) that

ηmax=ηmax<1  and then  ΔQW=ΔQWE36

We must be aware that for ηmax=ηmax<1the whole information entropy of the environment in which our (reversible) combined cycle OOis 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

H*(Y*)==ΔA*kTWH(X)ηmaxH(Y)ηmax=H(X)(ηmaxηmax)=0  E38

The whole combined machine or the thermodynamic system with the cycle OOis, when the cycle OOis 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 Oby the observing reverse process Owith 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 Aby 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 Ain volume Vand with nmatter units of ideal gas in the thermodynamic equilibrium. The state equation of Ais pV=nRΘ. For an elementary change of the internal energy Uof 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 δqbetween 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

S=n(cvdΘΘ+RdVV)=n(cvlnΘ+RlnV)+S0(n)=σ(Θ,V)+S0(n)  E39

Let us ‘divide’ the equilibrial system Ain a volume Vand 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 Asupposingly, to mparts Ai, i{1, …, m}, m1with volumes Viwith matter units ni. Evidently n=i=1mniand V=i=1mVi.

Let now S0(n)=0and S0i(ni)=0for all i. For the entropies Siof Aiconsidered individually, and for the change ΔS, when volumes V, Viare expressed from the state equations, and for p=pi, Θ=Θiit 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>0E40

Let us denote the last sum as Bfurther on, B<0. The quantity Bexpressed 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 mparts 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 Aby 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)=0is solvable for the system Aand all its parts Aiby 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 S0be the (change of) entropy ΔSwhich is added to the entropy σto figure out the measured entropy SClausof 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 σis 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.E42

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

Thus, our observation can be understood as an information transfer Tin an information channel Kwith 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.E43

For a number mof cells of our railings in the volume Vwith A, mNor for the accuracy rof this description of the ‘inner structure’ of A(a thought structure of Vwith A) and for the number qof 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.E44

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 ΔSagainst Sawaited, 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.


  • 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].
  • B. Russel, L. Whitehead, Principia Mathematica, 1910, 1912, 1913 and 1927.
  • 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 Refs. [3, 4, 5] and by means of all other, by them ‘called’, relations and functions (by their procedures).
  • And of the other relevant procedures too, see definitions 1–46 in Refs. [3, 4, 5].
  • We just think mistakenly that d≜ai+1 but ai+1=c is correct. Then the relation of Divisibility is not met. Neither is the relation of the Immediate Consequence.
  • Formulated in the language LP of the system P in compliance with its (with the TPA) inference rules.
  • πi is the i‐th prime number.
  • Caused by the application of the (Cantor) diagonal argument.
  • 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.
  • 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 [⋅⋅].
  • 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.
  • Thus it is not a common number as the [3, 4, 5] claims and neither is r.
  • In fact, it represents the very core of the sense of the Halting Problem task in the Computational Theory.
  • Attention (!) but x contains the message p that J(x)=J(x|p).
  • In the contrast to Refs. [3, 4, 5].
  • 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 ….’
  • 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.

© 2017 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Bohdan Hejna (December 20th 2017). Information Transfer and Thermodynamic Point of View on Goedel Proof, Ontology in Information Science, Ciza Thomas, IntechOpen, DOI: 10.5772/intechopen.68809. Available from:

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