Information Transfer and Thermodynamic Point of View on Goedel Proof

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.


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 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.

Goedel numbers, information and thermodynamics
Any inference within the system P 3 sets the T PA -theoretical relation 4 among its formulae a ½Á . This relation is given by their gradually generated special sequence a ! ¼ ½a 1 ; …; a q ; …; a p ; …; a k ; a kþ1 which is the proof of the latest inferred formula a kþ1 . By this, the unique arithmetic relation between their Goedel numbers, FORMULAE x ½Á , x ½Á ¼ Φða ½Á Þ, is set up, too. The gradually arising SEQUENCE of Þ is the PROOF of its latest FORMULA x kþ1 .
Let us assume that the given sequence a ! ¼ ½a o1 ; a o2 ; …; a o ; …; a q ; …; a p ; …; a k ; a kþ1 is a special one, and that, except of axioms (axiomatic schemes) a 01 ; …; a o , 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 a kþ1 of the proof a ! we use, from the a ! selected, (special) subsequence a q;p;kþ1 ! of the formulae a q ; a p ; a kþ1 .

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 ½a p i ;a q i ;a iþ1 ;½a p i ;a q i and ½a iþ1 with their information amounts Jða q i ;p i ;iþ1 ! Þ;Jða q i ;p i ! Þ and Jða iþ1 Þ. Index i is a serial number of the inferencing-analyzing-transferring step, The Goedel numbering also enables us to consider the individual Goedel numbers x i , x i jy i and y i of messages ½a p i ;a q i ;a iþ1 , ½a p i ;a q i a ½a iþ1 as messages too, with their (and the same) information amounts Jðx i Þ, Jðx i jy i Þ a Jðy i Þ, For each ith step of the goedelian syntactic analysis, we determine the values 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).
We regard these values as average values HðXÞ, HðXjYÞ and HðYÞ of information amounts of message sources X, XjY and Y with selective spaces X, X Â Y and Y, and with the uniform probability distribution, It is obvious that we consider a direct information transfer [11] through the channel K without noise, disturbing y i jx i , which means with the zero noise (disturbing) information ½Jðy i jx i Þ ¼ 0 In each ith step of the activity of our information model K of the arithmetic-syntactic analysis, it is valid that X : The relation Φða q i ;p i ;iþ1 ! ÞB Φða iþ1 Þ (x i B y i ) is evaluated by the relation of Divisibility and we identify its execution 7 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, And of the other relevant procedures too, see definitions 1-46 in Refs. [3][4][5]. of) a formula of the language L P of the system P. 8

Thermodynamic consideration
The thermodynamic consideration of an information transfer [11] reveals that the input message a q i ;p i ;iþ1 ! carries the input heat energy ΔQ Wi transformed by the reversible direct Carnot Cycle (Machine) C into the output mechanical work ΔA i corresponding to the output message a iþ1 . The heater A of the Carnot Cycle (Machine) C has the temperature T W and models the source of input messages (the message a q i ;p i ;iþ1 ! ) of the channel K. Its cooler B has the temperature T 0 determining the transfer efficiency η i . By the value η i > 0 the fact of inferrability of the chain a iþ1 from the special sequence of formulae a q i ;p i ;iþ1 ! as the formula of the theory T PA 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 L T P A of the theory T PA. 9 Thus, we have Now we obtain the information formulation [11] of the changes of the heat (thermodynamic) entropies ΔS C ½i , ΔS AB ½i and ΔS ½i A in the thermodynamic model C of our information transferinferring (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 analysisinformation transfer, the thermodynamic entropy S C of an isolated system, in which the modeling reversible direct Carnot Cycle C is running parallelly, increases in every ith step by the value ΔS C ½i , 8 We just think mistakenly that d ≜ a iþ1 but a iþ1 ¼ c is correct. Then the relation of Divisibility is not met. Neither is the relation of the Immediate Consequence. 9 Formulated in the language L P of the system P in compliance with its (with the T PA ) inference rules.
Provided that the ith inferring step has been done and written correctly (Modus Ponens) the Goedel arithmetic-syntactic analyzer decides, correctly, for the obtained a iþ1 ÞB Φða iþ1 Þ and Bew½Φða iþ1 Þ are valid, and the informationthermodynamic model ðK À CÞ generates the non-zero, positive output value TðX; YÞ for the Þ, respectively, and for Y : The zero change of the whole heat entropy S C 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 T PA , nothing is being inferred in the step i, ΔS C ½i ¼ 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 T PA -inference, we do not exert any 'useful effort' or energy in order to derive a new T PA -relation ½formula a iþ1 , FORMULA Φða iþ1 Þ. The previous 'effort' or energy associated with our inference (no matter that T PA -correct) of the sequence of a i i ! is worthless. The formula a iþ1 ½¼ d is just arbitrarily added to the previous sequence a i 1 ! of formulae of the theory T PA in such a way that it does not include any such formulae a q i and a p i that it would be valid Φða p i ;q i ;iþ1 ÞB Φða iþ1 Þ ¼ "1". In the information-thermodynamic interpretation, we write (for X : We have not exerted any inferring energy within the framework of building up the theory T PA , in order to create information Jðy i Þ > 0, and then we necessarily have η i ¼ 0; Jðy i Þ ¼ 0 where η i ¼ 0 expresses this error. All before a iþ1 , 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 a i included can be evaluated by the whole heat energy 10 10 π i is the i-th prime number.

Goedel substitution function and FORMULA 17Gen r
Let us consider the instance of the relation QðX;YÞ for the specific values x and y, X :¼ x and Y :¼ y, which is the constant relation Qðx;yÞ, and let us define the Goedel numbers y and y 0 that the Goedel (variable) number (his 'CLASS' SIGN) y arises from Admissible Substitution from the FORMULA qð17; 19Þ ½the ARITHMETIZATION of QðX;YÞ, Any of the following notations can be used The following Admissible Substitution Sb The CLAIM y 0 only seems to be a constant P/T PA -FORMULA, which, as the CLAIM y½ZðyÞ speaks only about a common number y. But, by the NUMERAL ZðyÞ 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 Type-1). In case of the 'global' validity of the substitution 19 :¼ ZðyÞ 11 it creates from the given FORMULA y an infinite sequence of semantically identical FORMULAE y 0 ½¼ y½ZðyÞ, y½Zðy 0 Þ ½¼ y½Z½y½ZðyÞ, … with the aim to end the process by 'printing' just the value y 0 . But it never reveals this outcome y 0 ; however, we -metatheoretically-know it. It never gets as far as to print the natural number y 0 which it 'wants to reach' by creating the infinite sequence of outcomes of the permanently repeated substitution 19 :¼ ZðyÞ which prevents it from this goal (y 0 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 11 Caused by the application of the (Cantor) diagonal argument.
analyzer is trying to calculate and 'print' y 0 , that prevents it from this aim. We never obtain a constant Goedel number. The FORMULA y½ZðyÞ arises by applying the (Cantor) diagonal argument, which is not any inference rule of the theory T PA (and of the system P), and thus, it is not an element of the language L T AP (and L P ). This is the reason for not-recursivity of the relations BewðÁÞ; the upper limit of its computing process is missing. First, we have q½ZðxÞ, 19 19 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 ½y½ZðyÞ ¼ 0 but, due to the inference properties, Neg½y½ZðyÞ ¼ 0 too. Then

19
yð19Þ 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 T PA -provability. The Goedel number y 0 , the FORMULA y½ZðyÞ, is seemingly a FORMULA (and even constant) of the system P and thus it is not an element of the theory T PA ; is not of an arithmetic type (it is not recursive arithmetically, only as for its basic syntax, syntactically). As the CLAIM y½ZðyÞ it speaks about the number y only, but by that it is the number y itself, then as y½ZðyÞ, it claims its own property, that from the Goedel number x it itself IS NOT INFERRED within the system P ½Bewðy 0 Þ ¼ 0. It is true for the given x and it 'says': 'I, FORMULA y½ZðyÞ, am in the system P 12 By substitution 19 :¼ ZðyÞ nothing changes in variability of FORMULA y 0 by the VARIABLE 19. The number y 0 should denote infinite and not recursive subset of natural numbers or to be equal to them.
(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 T PA .

FORMULA 17Gen r and information transfer
With regard of the fact that FORMULA y 0 is constructed by the diagonal argument, it is not INFERRED within the system P-in the T PA and so, it is not provable for any x from ℕ 0 . Then, within the framework of the theory T PA , we put 17Gen y 0 ¼ Def 0 and thus Jð17Gen y 0 Þ ¼ Def 0 . 13 In the proof we put p : 17Gen q, ½17 ffi u 1 ≜ X, 19 ≜ u 2 ≜ Y, q ¼ qð17, 19Þ, and then, in compliance with the Goedel notation, The metalanguage symbol QðX, YÞ in (20) or the symbol Qðℕ 0, YÞ is read as follows: The relation QðX;pÞ, QðX;pÞ ¼ ∀ x ∈ X jQ½x;Φ½∀ x ∈ X Qðx;pÞ and, therefore, the relation TðX;pÞ 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 T PA but gives a witness about it-about its property. It is so because it is formulated in a wider/general formulative language L PÃ than the language L P of the system P and so outside both of the language L P (and as such, outside of the language L T AP too). The FORMULAE/CLAIMS of both the theory T PA and the system P speak only about finite sets of arithmetic individuals but the theory T PA and the system P are the countable-N 0 -sets. 15 It seems only that 17Gen r is a part (of the ARITHMETIZATION) of the theory T PA 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 L PÃ in which all the arithmetic relations are written (and, in addition, the T PA -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 t½ZðxÞ;ZðyÞ be DESCRIPTION of the mechanism of the transfer y from x (on the level of the system P and the theory T PA ) in the channel K, 15 We have, inside of them, only N 0 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 N 1 containing, as its elements, the N 0 -sets; thus it can speak about the theory T PA , N 0 < N 1 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. A then the number yis not a FORMULA of the system P and in the information interpretation of inferring (INFERRING) within the system P it is valid that, JðyÞ ¼ 0 . Then we can consider the simultaneous validity of ½JðyÞ > 0&½JðyÞ < 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 0 ¼ p½ZðpÞ is valid So, the message p 0 (the message p about itself) is not-transferrable from any message x, ½xB ½K p 0 ¼ "1" ½xB ½K p ¼ "1" ½τ ½K ðx;yÞ ¼ "0" ½JðpÞ ¼ 0 ½Jðp 0 Þ ¼ 0 ð26Þ It is the attempt to transfer the message y (y ¼ 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 JðyÞ > 0 when it is applied in the (direct) transfer scheme K because the information JðyÞ > 0, y ¼ 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 T PA ) does not exist. In the theory, T PA is JðyÞ ¼ 0 for the CLAIM 17Gen r is not arithmetic at all, it is the metaarithmetic one. From the point of the theory T PA 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 a kþ1 ½CLAIM y, y ¼ Φða kþ1 Þ ¼ 17Gen r is not inferable (INFERABLE) in the given inferential system P (but in another one making its construction-INFERENCE possible, it is), 17 In fact, it represents the very core of the sense of the Halting Problem task in the Computational Theory. ∃ x ∈ X jt K ½JðxÞ;J½∃ x ∈ X t K ½JðxÞ;JðyÞ > 0 TðX;yÞ > 0 QðX;yÞ ð 27Þ 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 0 and on interrupted.' Or, we have made the interrupted transfer channel directly by this p 0 when we assumed it belonged to the set of messages transferrable from the source X. So, first we interrupt the channel K for p 0 , and then, we want to transfer this p 0 from the input x which includes this p 0 (or is identical to it), and so the internal and input state of the channel K are (also from the point of the theory T PA ) equivalent informationally. It is valid that Jðp 0 Þ ¼ 0 for any x, x ∈ X ½so ∀ c ∈ X j½Jðp 0 Þ ¼ 0, ∀ x ∈ X j½JðxÞ ¼ Jðxjp 0 Þ ffi ½JðXjp 0 Þ ¼ JðXÞ > 0 and for the simplicity is Jðp 0 jXÞ ¼ 0 The channel K, however, always works only with the not zero and the positive difference of information amounts JðxÞ À JðxjyÞ and in the theory T PA now it is valid that JðyÞ ¼ JðxÞÀ Jðxjp 0 Þ ¼ JðnullÞ ¼ 0, JðyÞ ¼ Jðp 0 Þ ¼ JðnullÞ ¼ 0 18 . It means that our assumption about p 0 ½¼ r is erroneous. No input message x having a relation to the output message p 0 exists. The FOR-MULA 17Gen r both creates and describes behavior of the not functioning (interrupted) information transfer, from p 0 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 xjy ½the information JðxjyÞ given as its input message x, it behaves as interrupted.
• There is no x ∈ ℕ 0 for which it is possible to generate the Goedel number Φ½Qðℕ 0 ;YÞ which claims that there is no x ∈ ℕ 0 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 ¼ Φða kþ1 Þ ¼ 17Gen r has not been INFERRED either. 18 Attention (!) but x contains the message p that JðxÞ ¼ JðxjpÞ . The metaarithmetic sense of the CLAIM ('SENTENTIAL PROPOSITION') 17Gen r is: • Within the general formulative language L PÃ of the inconsistent metasystem P * (containing the consistent subsystem P with the theory the T PA ) it is possible to construct ½be 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 T PA -they are the meta-T PA and the meta-P claims not belonging to the system P, but they belong to the inconsistent system P * , to its part P * À P (P * ⊃ 6 ¼ P).

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 T PA -arithmetic FORMULAE, is not such a FORMULA; it is not an element of the theory T PA and in convenience with [1,2,6,7,18,19] nor an element of the system P 19 and neither is r. The same is for Negð17Gen rÞ (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 VI 20 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, Negð17Gen rÞ and the Proposition VI as the claim about them are metaarithmetic (methodological) statements.

Auto-reference in information transfer, self-observation
In any information transfer channel K the channel equation HðXÞ À HðXjYÞ ¼ HðYÞ À HðYjXÞ ð 30Þ it is valid [?]. This equation describes the mutual relations among information entropies [(average) information amounts] in the channel K.
The quantities HðXÞ, HðYÞ, HðXjYÞ and HðYjXÞ are the input, the output, the loss and the noise entropy.
• For HðYjXÞ 6 ¼ 0 we have HðYÞ ¼ HðYjXÞ 6 ¼ 0 In both these two cases, the channel K operates as the interrupted (with the absolute noise) and the output HðYÞ is without any relation to the input HðXÞ and, also, it does not relate to the structure of K. This structure is expressed by the value of the quantity HðXjYÞ. We assume, for simplicity, that HðYjXÞ ¼ 0. 19 In the contrast to Refs. [3][4][5]. 20 Because, on the other hand, Goedel 1931 [3-5] 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 ….' 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, HðYÞ ¼ HðXÞ À HðXjYÞ > 0, is needed. (For more information see [14,15,16].

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 O ffi K is postulated. Further, we can imagine its observing method, equivalent to its 'mirror' O 00 ffi K 00 . This mirror O 00 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 00 in such a way that the mirror (the reverse cycle O 00 ) is gaining the message about the structure of the direct cycle O. This message is (carrying) the information HðXjYÞ about the structure of the transformation (transfer) process (O ffi K) being 'observed.' The mirror O 00 ffi K 00 is gaining this information HðXjYÞ on its noise 'input' HðY 0 0 jX 0 0 Þ [while HðX 0 0 Þ ¼ HðYÞ is its input entropy].
The quantities ΔQ W , ΔA and ΔQ 0 or the quantities ΔQ 00 W , ΔA 0 0 and ΔQ 0 0 0 , respectively, define the information entropies of the information transfer realized (thermodynamically) by the direct Carnot Cycle O or by the reverse Carnot Cycle O 00 (the mirror), respectively, (the combined cycle OO 00 is created), Our aim is to gain the non-zero output mechanical work ΔA Ã of the combined heat cycle OO 00 , To achieve this aim, for the efficiencies η max and η 00 max of the both connected cycles O and O 00 (with the working temperatures T W ¼ T 0 0 W and T 0 ¼ T 0 0 0 , T W ≥ T 0 > 0), it must be valid that η max > η 00 max ; we want the validity of the relation 21 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.
When ΔQ 0 ¼ ΔQ 00 0 should be valid, then must be that ΔQ 00 W < ΔQ W ½( ðη max > η 00 max Þ, and thus, it should be valid that Thus, the output work ΔA Ã > 0 should be generated without any lost heat and by the direct change of the whole heat ΔQ W À ΔQ 00 W but within the cycle OO 00 . For η max < η 00 max the same heat ΔQ W À ΔQ 00 W should be pumped from the cooler with the temperature T 0 to the heater with the temperature T W directly, without any compensation by a mechanical work. We see that ΔA Ã ¼ 0 is the reality. Our combined machine OO 00 should be the II: Perpetuum Mobile in both two cases. Thus, η max ¼ η 00 max must be valid (the heater with the temperature T W and the cooler with the temperature T 0 are common) that We must be aware that for η max ¼ η 00 max < 1 the whole information entropy of the environment in which our (reversible) combined cycle OO 00 is running changes on one hand by the value and on the other hand it is also changed by the value ÀHðXÞ Á η max ¼ À ΔQ W kTW Á ð1 À βÞ Thus, it must be changed by the zero value The whole combined machine or the thermodynamic system with the cycle OO 00 is, when the cycle OO 00 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 00 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].

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 ¼ nRΘ. For an elementary change of the internal energy U of A, we have dU ¼ nc v dΘ.
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 ¼ dU þ pdV From this principle, and from Clausius equation 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 A i , i ∈ {1; …; m}, m ≥ 1 with volumes V i with matter units n i . Evidently n ¼ Let now S 0 ðnÞ ¼ 0 and S 0i ðn i Þ ¼ 0 for all i. For the entropies S i of A i considered individually, and for the change ΔS, when volumes V; V i are expressed from the state equations, and for p ¼ p i , Θ ¼ Θ i it will be gained that σ ½i ¼ Rn ½i lnn ½i . Then, for Let us denote the last sum as B further on, B < 0. The quantity ÀB expressed in (40) is information entropy of a source of messages with an alphabet ½n 1 ; n 2 ; … ;n m and probability . Such a division of the system to m parts defines an information source with the information entropy with its maximum ln m.
The result (37), ΔS ¼ ÀnRB, 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 S 0 ðnÞ, S 0i ðn i Þ, in such a way, that the equation ΔS ¼ ðσ þ S 0 Þ À X m i¼1 ðσ i þ S 0i Þ ¼ 0 is solvable for the system A and all its parts A i by solutions S 0½i ðn ½i Þ ¼ Àn ½i R ln n ½i γ ½i .
Then, S ½i ≜ S Claus ½i , and we write and derive that Now let us observe an equilibrium, S * ¼ S Claus ¼ S Boltz ¼ ÀkNBÃ ¼ ÀkN ln N.
Let, in compliance with the solution of Gibbs Paradox, the integration constant S 0 be the (change of) entropy ΔS which is added to the entropy σ to figure out the measured entropy S Claus 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 ¼ S Claus À ΔS; ΔS ¼ S 0 .
Following the previous definitions and results, we have By the entropy ΔS the 'lost' heat ΔQ 0 (at the temperature Θ) is defined.
Thus, our observation can be understood as an information transfer T in an information channel K with entropies HðXÞ, HðYÞ, HðXjYÞ and HðYjXÞ in (33) but now bound physically; we have these information entropies per one particle of the observed system A: For a number m of cells of our railings in the volume V with A, m ≤ N 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; m À 1 >, we have that r ¼ N À 1 q .
Our observation of the equilibrium system A, including the mathematical correction for Gibbs Paradox, is then describable by the Shannon transfer scheme ½X; K; Y , where However, a real observation process described in (44), equivalent to that one with r ¼ 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 ½ΔS < 0 & ½ΔS > 0-see the validity of the Goedel Proposition V [3-5] 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).