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
- information transfer
- information entropy
- heat efficiency
- infinite cycle
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 . 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 3 sets the ‐theoretical relation4 among its formulae . This relation is given by their gradually generated special sequence which is the proof of the latest inferred formula . By this, the unique arithmetic relation between their Goedel numbers,
Let us assume that the given sequence is a special one, and that, except of axioms (axiomatic schemes) , 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 of the proof we use, from the selected, (special) subsequence of the formulae . The formulae , have already been derived, or they are axioms. It is valid that , and we assume that ,
Checking the syntactic and TPA‐theoretical correctness of the analyzed chains , as the formulae of the system having been generated by inferring (Modus Ponens) within the system (in the theory ), and also the special sequence of the formulae of the system (theory ), is realized by checking the arithmetic‐syntactic correctness of the notation of their corresponding FORMULAE and SEQUENCE of FORMULAE, by means of the relations ‘called’ from (the sequence of procedures) relations 6; the core of the whole (goedelian) arithmetic‐syntactic analysis is the (procedure) relation of Divisibility,
2.1. Inference in the system and information transfer
The syntactic analysis of the special sequence of the formulae of the system 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 .
As such, it is a sequence of successive attempts to transfer information with input, loss and output messages with their information amounts . Index is a serial number of the inferencing—analyzing—transferring step, . The Goedel numbering also enables us to consider the individual Goedel numbers , and of messages , a as messages too, with their (and the same) information amounts , a ,
For each ith step of the goedelian syntactic analysis, we determine the values
We regard these values as average values , and of information amounts of message sources , and with selective spaces , and , and with the uniform probability distribution,
It is obvious that we consider a direct information transfer  through the channel without noise, disturbing , which means with the zero noise (disturbing) information .
In each ith step of the activity of our information model of the arithmetic‐syntactic analysis, it is valid that and , and the channel equation is applicable ,
The relation () is evaluated by the relation of Divisibility and we identify its execution7 with the actual direct information transfer in the channel . So, when our inference by Modus Ponens is done correctly, in each ith step, we have its information interpretation, in steps i,
Let us assume that, when inferring by Modus Ponens, , we make such an error that we write , where, however, the chain (by chance) can also be (in the form of) a formula of the language of the system .8 For the considered NOT‐INFERRABILITY of , being interpreted now from the point of information view, we put , or better said, with regard of the properties of INFERENCE, we are forced to put within the framework of the theory and then, informationally
2.2. Thermodynamic consideration
The thermodynamic consideration of an information transfer  reveals that the input message carries the input heat energy transformed by the reversible direct Carnot Cycle (Machine) into the output mechanical work corresponding to the output message . The heater of the Carnot Cycle (Machine) has the temperature and models the source of input messages (the message ) of the channel . Its cooler has the temperature determining the transfer efficiency . By the value the fact of inferrability of the chain from the special sequence of formulae as the formula of the theory is stated.
Thus, the reversible direct Carnot Cycle is the thermodynamic model of the direct information transfer through the channel , and hereby of the inferring (INFERRING) itself, and also of the arithmetic‐syntactic analysis of formulae of the language of the theory .8 Thus, we have
Now we obtain the information formulation  of the changes of the heat (thermodynamic) entropies , and in the thermodynamic model of our information transfer—inferring (INFERRING)—arithmetic‐syntactic analysis within the (language of the) system ,
In accordance with Ref. , it is valid that, within the inferring—arithmetic‐syntactic analysis—information transfer, the thermodynamic entropy of an isolated system, in which the modeling reversible direct Carnot Cycle is running parallelly, increases in every ith step by the value ,
Provided that the ith inferring step has been done and written correctly (Modus Ponens) the Goedel arithmetic‐syntactic analyzer decides, correctly, for the obtained , that the relations and are valid, and the information‐thermodynamic model generates the non‐zero, positive output value for the inferring step [for or , respectively, and for ],
The zero change of the whole heat entropy of the isolated system in which our model cycle is running occurs just when in the inferential system , from the perspective of the theory TPA, nothing is being inferred in the step , . 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 ‐inference, we do not exert any ‘useful effort’ or energy in order to derive a new ‐relation formula , FORMULA . The previous ‘effort’ or energy associated with our inference (no matter that ‐correct) of the sequence of is worthless. The formula is just arbitrarily added to the previous sequence of formulae of the theory in such a way that it does not include any such formulae and that it would be valid . In the information‐thermodynamic interpretation, we write (for )
We have not exerted any inferring energy within the framework of building up the theory , in order to create information , and then we necessarily have where expresses this error. All before , otherwise inferred correctly, is not related to it–the information transfer channel is interrupted. The overall amount of our inference efforts exerted in vain up to included can be evaluated by the whole heat energy10
3. Goedel substitution function and
Let us consider the instance of the relation for the specific values and , and , which is the constant relation , and let us define the Goedel numbers and that the Goedel (variable) number (his ‘CLASS’ SIGN) arises from Admissible Substitution from the FORMULA the
Any of the following notations can be used
The following Admissible Substitution is carried out in the second step of the given Double Substitution ; in the Goedel variable number , we first put and in the result we put . Then
The CLAIM only seems to be a constant /‐FORMULA, which, as the CLAIM speaks only about a common number . But, by the NUMERAL it is the speaking about and then, it is the FORMULA 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‐). In case of the ‘global’ validity of the substitution 11 it creates from the given FORMULA an infinite sequence of semantically identical FORMULAE , , with the aim to end the process by ‘printing’ just the value . But it never reveals this outcome ; however, we—metatheoretically—know it. It never gets as far as to print the natural number which it ‘wants to reach’ by creating the infinite sequence of outcomes of the permanently repeated substitution which prevents it from this goal (marks the claim about the claim , the claim about the claim about the claim etc.). It is even the first one, by which the analyzer is trying to calculate and ‘print’ , that prevents it from this aim. We never obtain a constant Goedel number. The FORMULA arises by applying the (Cantor) diagonal argument, which is not any inference rule of the theory (and of the system ), and thus, it is not an element of the language (and ). This is the reason for not‐recursivity of the relations ; the upper limit of its computing process is missing. First, we have 12
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 but, due to the inference properties, too. Then
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 ‐provability. The Goedel number , the FORMULA , is seemingly a FORMULA (and even constant) of the system and thus it is not an element of the theory ; is not of an arithmetic type (it is not recursive arithmetically, only as for its basic syntax, syntactically). As the CLAIM it speaks about the number only, but by that it is the number itself, then as , it claims its own property, that from the Goedel number it itself IS NOT INFERRED within the system . It is true for the given and it ‘says’: ‘I, FORMULA , am in the system (by it means) from the Goedel number UNPROVABLE.’ And, by this, it also states both the property of the system and the theory .
3.1. FORMULA and information transfer
With regard of the fact that FORMULA is constructed by the diagonal argument, it is not INFERRED within the system —in the and so, it is not provable for any from . Then, within the framework of the theory , we put and thus .13 In the proof we put , , and then, in compliance with the Goedel notation,
The metalanguage symbol in (20) or the symbol is read as follows:
‘None is in the relation INFERENCE to the content (to the selective space ) of the variable . From any given , , any Goedel number , writable as the proposed outcome of the INFERENCE from the given , is NOT INFERRED in reality.’
We put .
The Goedel number r, is, by the substitution , 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, ,
Further14 and then,
With regard of quantification over values of the variable , we write
The relation , and, therefore, the relation says that no such exists to comply with the message transfer conditions of from ; the infinite cycle is stipulated. Attempts to give the proof of the FORMULA within the framework of the inferential system , that is, attempts to ‘decide’ it inside the system only by the means of the system itself end up in the infinite cycle.
The claim does not belong to the theory but gives a witness about it—about its property. It is so because it is formulated in a wider/general formulative language than the language of the system and so outside both of the language (and as such, outside of the language too). The FORMULAE/CLAIMS of both the theory and the system speak only about finite sets of arithmetic individuals but the theory and the system are the countable–‐sets.15 It seems only that is a part (of the ARITHMETIZATION) of the theory and of the system which is by it is written down (grammatically only) according to the common/general recursive syntax of the general formulative language in which all the arithmetic relations are written (and, in addition, the ‐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 be DESCRIPTION of the mechanism of the transfer from (on the level of the system and the theory ) in the channel ,
But, when it is valid that then the number is not a FORMULA of the system and in the information interpretation of inferring (INFERRING) within the system it is valid that, . Then we can consider the simultaneous validity of —also see the Proposition 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 and Goedel number is valid
So, the message (the message about itself) is not‐transferrable from any message ,
It is the attempt to transfer the message () through the channel , while this message itself causes its interruption and ‘wants’ to be transferred through this interrupted channel as well.17 Its ‘errorness’ is in our awaiting of the non‐zero outcome when it is applied in the (direct) transfer scheme because the information , (known from and valid in the metalanguage), from the point of transferrability through the channel (from the point of inferrability in the theory ) does not exist. In the theory, TPA is for the CLAIM is not arithmetic at all, it is the metaarithmetic one. From the point of the theory and the system , it is not quite well to call CLAIM as the SENTENTIAL FORMULA; it has only such form. For this reason, we use the term CLAIM or ‘SENTENTIAL FORMULA’/‘PROPOSITION.’
The message about that the channel is for interrupted cannot be transferred through the same channel interrupted for (however, through another one, uninterrupted for , it can). Or we can say that the claim CLAIM , is not inferable (INFERABLE) in the given inferential system (but in another one making its construction‐INFERENCE possible, it is),
By constructing the FORMULA and from the point of information transfer, we have produced the claim ‘the transfer channel is from and on interrupted.’ Or, we have made the interrupted transfer channel directly by this when we assumed it belonged to the set of messages transferrable from the source . So, first we interrupt the channel for , and then, we want to transfer this from the input which includes this (or is identical to it), and so the internal and input state of the channel are (also from the point of the theory ) equivalent informationally. It is valid that for any , so ,
The channel , however, always works only with the not zero and the positive difference of information amounts and in the theory now it is valid that , 18. It means that our assumption about is erroneous. No input message having a relation to the output message exists. The FORMULA both creates and describes behavior of the not functioning (interrupted) information transfer, from on further. For the efficiency of the information transfer, it is then valid [14, 16] that
The CLAIM (‘SENTENTIAL PROPOSITION’) we interpret as follows:
No information transfer channel transfers its (internal) state the information given as its input message , it behaves as interrupted.
There is no for which it is possible to generate the Goedel number which claims that there is no for which it is possible to generate the non‐zero Goedel number that we could write into the variable . This means that from any Goedel number no INFERENCE is possible just for its latest part has not been INFERRED either.
The metaarithmetic sense of the CLAIM (‘SENTENTIAL PROPOSITION’) is:
Within the general formulative language of the inconsistent metasystem (containing the consistent subsystem with the theory the ) it is possible to construct be the (Cantor) diagonal argument such a claim (with the Goedel arithmetization code ) which is true, but both this claim i and its negation are not provable/PROVABLE by the means of the system (in the system ) and thus, also in the theory –they are the meta‐and the meta‐claims not belonging to the system , but they belong to the inconsistent system , to its part ().
So, the Goedel Proposition (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 .’
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’ , no matter how much it complies with the grammar of recursive writing of ‐arithmetic FORMULAE, is not such a FORMULA; it is not an element of the theory and in convenience with [1, 2, 6, 7, 18, 19] nor an element of the system 19 and neither is . The same is for (it cannot be inferred in for is not inferable in .) Nevertheless, we are in accordance with the intuitive and obviously intended sense of the Goedel Proposition 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’) , and the Proposition as the claim about them are metaarithmetic (methodological) statements.
5.1. Auto-reference in information transfer, self-observation
In any information transfer channel the channel equation
it is valid [?]. This equation describes the mutual relations among information entropies [(average) information amounts] in the channel .
The quantities , , and are the input, the output, the loss and the noise entropy.
The difference or the difference defines the transinformation or the transinformation , respectively,
When the channel transfers the information (entropy) , but now just at the value of the entropy , , then, necessarily, must be valid
For , we have .
For we have
In both these two cases, the channel operates as the interrupted (with the absolute noise) and the output is without any relation to the input and, also, it does not relate to the structure of . This structure is expressed by the value of the quantity . We assume, for simplicity, that .
From Eqs. (30) to (32) follows that the channel cannot transfer (within the same step 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 cannot transfer its own states considered as the input messages (within the same steps ). 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, , 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 is possible to be comprehended (modeled or, even, constructed) as the direct Carnot Cycle [8, 10]. The relation is postulated. Further, we can imagine its observing method, equivalent to its ‘mirror’ . This mirror is, at this case, the direct Carnot Cycle as for its structure, but functioning in the indirect, reverse mode [8, 10].
Let us connect them together to a combined heat cycle in such a way that the mirror (the reverse cycle ) is gaining the message about the structure of the direct cycle . This message is (carrying) the information about the structure of the transformation (transfer) process () being ‘observed.’ The mirror is gaining this information on its noise ‘input’ [while is its input entropy].
The quantities , and or the quantities , and , respectively, define the information entropies of the information transfer realized (thermodynamically) by the direct Carnot Cycle or by the reverse Carnot Cycle (the mirror), respectively, (the combined cycle is created),
Our aim is to gain the non‐zero output mechanical work of the combined heat cycle , . We want to gain non‐zero information .
To achieve this aim, for the efficiencies and of the both connected cycles and (with the working temperatures and , ), it must be valid that ; we want the validity of the relation21
When should be valid, then must be that , and thus, it should be valid that
Thus, the output work should be generated without any lost heat and by the direct change of the whole heat but within the cycle . For the same heat should be pumped from the cooler with the temperature to the heater with the temperature directly, without any compensation by a mechanical work. We see that is the reality.
Our combined machine should be the Perpetuum Mobile in both two cases. Thus, must be valid (the heater with the temperature and the cooler with the temperature are common) that
We must be aware that for the whole information entropy of the environment in which our (reversible) combined cycle is running changes on one hand by the value
and on the other hand it is also changed by the value Thus, it must be changed by the zero value
The whole combined machine or the thermodynamic system with the cycle is, when the cycle 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 by the observing reverse process 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 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 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 in volume and with matter units of ideal gas in the thermodynamic equilibrium. The state equation of is . For an elementary change of the internal energy of , we have .
From the state equation of , and from the general law of energy conservation [for a (substitute) reversible exchange of heat between the system and its environment], we formulate the Principle of Thermodynamics,
From this principle, and from Clausius equation , , follows that
Let us ‘divide’ the equilibrial system in a volume and at a temperature , or, better said, the whole volume (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 supposingly, to parts , , with volumes with matter units . Evidently and .
Let now and for all . For the entropies of considered individually, and for the change , when volumes are expressed from the state equations, and for , it will be gained that . Then, for is valid, we have that
Let us denote the last sum as further on, . The quantity expressed in (40) is information entropy of a source of messages with an alphabet and probability distribution . Such a division of the system to parts defines an information source with the information entropy with its maximum .
The result (37), , is a paradox, a contradiction with our presumption of not influencing a thermodynamic state of by diaphragms, and, leads to that result that the heat entropy (of a system in equilibrium) is not an extensive quantity. But, by the definition of the differential , this is not true.
Due to this contradiction, we must consider a non‐zero integrating constants , , in such a way, that the equation is solvable for the system and all its parts by solutions .
Then, , and we write and derive that
Now let us observe an equilibrium, .
Let, in compliance with the solution of Gibbs Paradox, the integration constant be the (change of) entropy which is added to the entropy to figure out the measured entropy of the equilibrium state of the system (the final state of Gay‐Lussac experiment) at a temperature . We have shown that without such correction, the less entropy is evidenced, .
Following the previous definitions and results, we have
By the entropy the ‘lost’ heat (at the temperature ) is defined.
Thus, our observation can be understood as an information transfer in an information channel with entropies , , and in (33) but now bound physically; we have these information entropies per one particle of the observed system :
For a number of cells of our railings in the volume with , or for the accuracy of this description of the ‘inner structure’ of (a thought structure of with ) and for the number of diaphragms creating our railings of cells and constructed in such a way that , we have that .
Our observation of the equilibrium system , including the mathematical correction for Gibbs Paradox, is then describable by the Shannon transfer scheme , where
However, a real observation process described in (44), equivalent to that one with , is impossible.
We conclude by that, the diminishing of the measured entropy value about against 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 (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 –see the validity of the Goedel Proposition [3, 4, 5] for the inconsistent system .
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).
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. , 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.
- From that y′ is NOT INFERRED follows its NOT‐INFERRABILITY/NOT‐PROVABILITY.
- 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 , of the I. type.