The choice of communication channels through which agricultural knowledge flowed was influenced by several factors. Among famers, availability and accessibility of the communication channels, level of development of ICT infrastructure, affordability of tariffs for communication and ownership of some communication tools were found to influence the choice of channels used for either sharing or exchanging agricultural knowledge. Likewise, among non-farmers agricultural stakeholders access to ICT infrastructure and availability and accessibility of ICT tools, affordability of tariffs, availability, convenience and suitability of communication channels for a communication process influence the choice of communication channels.

Without considering other factors, people use communication channels which are available and easily accessible [39]. This applies to ICT- and non-ICT-based channels. For ICT-based channels, the availability and accessibility of a channel is explained by the level of development of ICT infrastructure and accessibility of ICT tools. Since the flow of knowledge involves some costs, then the affordability of costs associated with a specific communication channel influences the choice of communication channels. When those sharing agricultural knowledge afford costs associated with the process, then affordable communication channels are more likely to be used [40]. Likewise, those who afford to own communication tools are more likely to use such tools for enhancing knowledge flow. For example, those owning agricultural books are more likely to read them than non-owners. Also, those owning ICT tools are more likely to use for either sharing or exchanging agricultural knowledge than for non-owners. Moreover, communication channels, which are more convenient and suitable for a knowledge flow process, are more likely to be used for either knowledge sharing, exchange or dissemination.

## 5.3. Factors influencing the flow of agricultural knowledge among stakeholders

There are several factors known to influence the flow of agricultural knowledge among agricultural stakeholders. Membership in farmers'/professional group and accessibility of agricultural knowledge sources influence the flow of agricultural knowledge. The flow of agricultural knowledge is known to be higher among farmers found in groups [41] because each farmer can be either a knowledge source or a recipient. Moreover, providers of agricultural knowledge services find it easy to reach more farmers or other actors when they were in groups than as individuals [41].

To enhance agricultural knowledge flow, communities and organizations should cultivate a knowledge-sharing culture. Community/organizational culture is expressed in terms of leadership, sociability, solidarity, trust, core beliefs, values, norms and social customs [44]. When all these elements of community/organizational culture support the creation and sharing of knowledge, then the level of knowledge accessibility becomes high. Communities and organizations should create platforms through which members can share and exchange knowledge among them. In agricultural organizations, the top management has a strong role to play so as to enhance knowledge-sharing culture, hence increasing the flow of agricultural knowledge. The top management in agricultural institutions can create a rewarding system so as to promote knowledge sharing, exchange and dissemination. This in turn increases the level of

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273

Equally, the availability, convenience, suitability of communication channels to sharing and exchanging agricultural knowledge may influence the level of flow of agricultural knowledge. Agricultural stakeholders use the most available, convenient and suitable communication channels for sharing, exchanging or disseminating agricultural knowledge [39]. This in turn

For agricultural information services providers, the affordability of taxes paid for their services and the availability of sponsorship play an important role in increasing the rate of dissemination of agricultural to mass. Governments exert some taxes or duties to media houses. When such taxes or duties are too high, the operational cost of such houses becomes higher limiting them from disseminating agricultural knowledge. Sponsors are important for cushioning the impacts of high taxes and duties on disseminating agricultural knowledge. However, this is

Findings also show that there is correlation between demographic characteristics and some agricultural knowledge flow processes. There is negative correlation between reporting observed farm-related problem and farmer's level of education (r = 0.302(\*\*) at 0.000 level of significance). This indicates that as farmer's level of education increases, the ability to handle problems found at farms increases. Likewise, findings indicate that there is a negative correlation between accessing agricultural knowledge and farmer's level of education (r = 0.194(\*\*) at 0.001 level of significance). This tells that as the farmer's level of education increased, the level of accessing agricultural knowledge declined. In other words, educated farmers thought they had most of the knowledge needed for production that accessing it from a third party was not important. Findings also indicate that there is a significant positive correlation between accessing agricultural knowledge and farmer's age (r = 0.203 (\*\*) at 0.000 level of significance) and farming experience (r = 0.138(\*) at 0.014 level of significance). This tells that the level of accessing agricultural knowledge is higher among old farmers than young ones. In other words, old farmers enhance more flow of agricultural

An effective agricultural knowledge flow is important for increased accessibility, usage and creation of knowledge, hence improving agricultural productivity. Agricultural knowledge

flow of agricultural knowledge.

increases the level of flow of agricultural knowledge.

only possible when sponsorship is available.

knowledge than young ones.

5.4. Conclusion and recommendations

Likewise, the affordability of tariffs for communication influences the flow of agricultural knowledge. Some knowledge flow processes are not free of charge. The recipient or the one disseminating knowledge has to pay some fee so as to communicate knowledge. In most cases, it is only when such costs are low and affordable then the knowledge flow process becomes high. Affordability can equally relate to one's ability to own communication tools too. If communication tools are owned, then the level of usage of such tools among owners becomes higher, hence increasing the possibilities of using them for sharing or exchanging agricultural knowledge.

Among farmers, agricultural extension staff play an important role in enhancing access to knowledge. They are designed to build and strengthen the capacity of rural farmers and other stakeholders through enhancing access to knowledge [42]. To easily reach more stakeholders and enhance access to agricultural knowledge among them, agricultural extension staff should have access to transport facilities. Transport facilities help agricultural extension staff meet more stakeholders within limited time and hence disseminating or exchanging knowledge with more stakeholders.

Other factors, availability of agricultural knowledge, having a well-developed ICT infrastructure and a reliable power sources and airing agricultural radio/TV programs during relevant hours influence the flow of agricultural knowledge. Agricultural knowledge can only flow from one point to the other when it is available. Reliable sources of power are important for ICT-based channels. Available agricultural knowledge can only be shared, exchanged or disseminated through ICTs if such tools are connected to a source of power. Radio and TV sets, computers and mobile phones are among the tools that can only work when connected to a source of power. When broadcasting agricultural knowledge through radio and TV sets, it is important to consider the relevance of time to the target audience of the radio/TV agricultural programs. Without time consideration, few or none of the intended audience can access broadcasted contents. Moreover, a well-developed ICT infrastructure is important for such broadcast to reach more of the intended audience. Likewise, wide wired and wireless phone connections are important for enhancing the flow of available agricultural knowledge among agricultural stakeholders.

Likewise, the usage of most suitable language for repackaging knowledge influences the level of flow of agricultural knowledge. Using unknown or foreign language to repackaging agricultural knowledge limits some stakeholders from accessing knowledge [43]. Very important agricultural knowledge found in an unknown language will not be accessed, shared or exchanged among stakeholders. Therefore, repackaging knowledge in a-not-known and difficult language is like burying it.

An efficient feedback mechanism is important for increasing the flow of agricultural knowledge among agricultural stakeholders. Communication channels enhancing immediate feedback are preferred more than those that do not [39]. Moreover, such channels are more likely to enhance effective agricultural knowledge flow among stakeholders.

To enhance agricultural knowledge flow, communities and organizations should cultivate a knowledge-sharing culture. Community/organizational culture is expressed in terms of leadership, sociability, solidarity, trust, core beliefs, values, norms and social customs [44]. When all these elements of community/organizational culture support the creation and sharing of knowledge, then the level of knowledge accessibility becomes high. Communities and organizations should create platforms through which members can share and exchange knowledge among them. In agricultural organizations, the top management has a strong role to play so as to enhance knowledge-sharing culture, hence increasing the flow of agricultural knowledge. The top management in agricultural institutions can create a rewarding system so as to promote knowledge sharing, exchange and dissemination. This in turn increases the level of flow of agricultural knowledge.

Equally, the availability, convenience, suitability of communication channels to sharing and exchanging agricultural knowledge may influence the level of flow of agricultural knowledge. Agricultural stakeholders use the most available, convenient and suitable communication channels for sharing, exchanging or disseminating agricultural knowledge [39]. This in turn increases the level of flow of agricultural knowledge.

For agricultural information services providers, the affordability of taxes paid for their services and the availability of sponsorship play an important role in increasing the rate of dissemination of agricultural to mass. Governments exert some taxes or duties to media houses. When such taxes or duties are too high, the operational cost of such houses becomes higher limiting them from disseminating agricultural knowledge. Sponsors are important for cushioning the impacts of high taxes and duties on disseminating agricultural knowledge. However, this is only possible when sponsorship is available.

Findings also show that there is correlation between demographic characteristics and some agricultural knowledge flow processes. There is negative correlation between reporting observed farm-related problem and farmer's level of education (r = 0.302(\*\*) at 0.000 level of significance). This indicates that as farmer's level of education increases, the ability to handle problems found at farms increases. Likewise, findings indicate that there is a negative correlation between accessing agricultural knowledge and farmer's level of education (r = 0.194(\*\*) at 0.001 level of significance). This tells that as the farmer's level of education increased, the level of accessing agricultural knowledge declined. In other words, educated farmers thought they had most of the knowledge needed for production that accessing it from a third party was not important. Findings also indicate that there is a significant positive correlation between accessing agricultural knowledge and farmer's age (r = 0.203 (\*\*) at 0.000 level of significance) and farming experience (r = 0.138(\*) at 0.014 level of significance). This tells that the level of accessing agricultural knowledge is higher among old farmers than young ones. In other words, old farmers enhance more flow of agricultural knowledge than young ones.

## 5.4. Conclusion and recommendations

can be either a knowledge source or a recipient. Moreover, providers of agricultural knowledge services find it easy to reach more farmers or other actors when they were in groups than

Likewise, the affordability of tariffs for communication influences the flow of agricultural knowledge. Some knowledge flow processes are not free of charge. The recipient or the one disseminating knowledge has to pay some fee so as to communicate knowledge. In most cases, it is only when such costs are low and affordable then the knowledge flow process becomes high. Affordability can equally relate to one's ability to own communication tools too. If communication tools are owned, then the level of usage of such tools among owners becomes higher, hence increasing

Among farmers, agricultural extension staff play an important role in enhancing access to knowledge. They are designed to build and strengthen the capacity of rural farmers and other stakeholders through enhancing access to knowledge [42]. To easily reach more stakeholders and enhance access to agricultural knowledge among them, agricultural extension staff should have access to transport facilities. Transport facilities help agricultural extension staff meet more stakeholders within limited time and hence disseminating or exchanging knowledge

Other factors, availability of agricultural knowledge, having a well-developed ICT infrastructure and a reliable power sources and airing agricultural radio/TV programs during relevant hours influence the flow of agricultural knowledge. Agricultural knowledge can only flow from one point to the other when it is available. Reliable sources of power are important for ICT-based channels. Available agricultural knowledge can only be shared, exchanged or disseminated through ICTs if such tools are connected to a source of power. Radio and TV sets, computers and mobile phones are among the tools that can only work when connected to a source of power. When broadcasting agricultural knowledge through radio and TV sets, it is important to consider the relevance of time to the target audience of the radio/TV agricultural programs. Without time consideration, few or none of the intended audience can access broadcasted contents. Moreover, a well-developed ICT infrastructure is important for such broadcast to reach more of the intended audience. Likewise, wide wired and wireless phone connections are important for enhancing the flow of available agricultural knowledge among

Likewise, the usage of most suitable language for repackaging knowledge influences the level of flow of agricultural knowledge. Using unknown or foreign language to repackaging agricultural knowledge limits some stakeholders from accessing knowledge [43]. Very important agricultural knowledge found in an unknown language will not be accessed, shared or exchanged among stakeholders. Therefore, repackaging knowledge in a-not-known and diffi-

An efficient feedback mechanism is important for increasing the flow of agricultural knowledge among agricultural stakeholders. Communication channels enhancing immediate feedback are preferred more than those that do not [39]. Moreover, such channels are more likely to

enhance effective agricultural knowledge flow among stakeholders.

the possibilities of using them for sharing or exchanging agricultural knowledge.

as individuals [41].

272 Ontology in Information Science

with more stakeholders.

agricultural stakeholders.

cult language is like burying it.

An effective agricultural knowledge flow is important for increased accessibility, usage and creation of knowledge, hence improving agricultural productivity. Agricultural knowledge

sharing, exchange, transfer and dissemination are important processes facilitating the flow of agricultural knowledge among agricultural stakeholders. For these processes to be effective, suitable and convenient communication channels should be available to enhance the flow of knowledge from the source to the recipient. Paper-based, human- and ICT-based communication channels are commonly used for the flow of agricultural knowledge. Availability of agricultural knowledge and accessibility of knowledge sources are important for enhanced agricultural flow. Well-developed ICT infrastructure, access to power sources and an effective feedback mechanism play an important role in enhancing adequate flow of agricultural knowledge. For agricultural knowledge to be understandable, sharable and exchangeable, it should be repackaged using appropriate languages. Communities and organizations should have strategies to enhance a continuous flow of agricultural knowledge among members. In order to improve the flow of agricultural knowledge, the involvement of different stakeholders is inevitable. Each agricultural stakeholder should be involved in conducting relevant agricultural knowledge-related roles so as to enhance the accessibility, sharing, exchange, dissemination and usage of agricultural knowledge. In order to increase the flow of agricultural knowledge, the Government in partnership with the private sector should widen the ICT and agricultural communication infrastructure.

[4] Isinika AC. Evaluating agricultural research and extension in Tanzania: The production function approach. In: Anandajayasekeram P, Rukuni M, Babu S, Liebenberg F, Keswani CL, editors. Impact of Science on African Agriculture and Food Security. King's Lynn: CABI;

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2007:127-136

ehost-live

j.hrmr.2009.10.001

<EMAIL_ADDRESS>
Systems. Naples, Italy; 2013

DOI: https://doi.org/10.1016/j.iilr.2010.07.006

DOI: https://doi.org/10.1108/14684520910951186

1079. DOI: https://doi.org/10.1016/j.jbusres.2006.12.014

agement and E-Learning. 2009;1(2):139-162

https://doi.org/10.1177/0165551504046997

## Author details

Wulystan P. Mtega1 \* and Mpho Ngoepe<sup>2</sup>

\*Address all correspondence to<EMAIL_ADDRESS>

## References

[4] Isinika AC. Evaluating agricultural research and extension in Tanzania: The production function approach. In: Anandajayasekeram P, Rukuni M, Babu S, Liebenberg F, Keswani CL, editors. Impact of Science on African Agriculture and Food Security. King's Lynn: CABI; 2007:127-136

sharing, exchange, transfer and dissemination are important processes facilitating the flow of agricultural knowledge among agricultural stakeholders. For these processes to be effective, suitable and convenient communication channels should be available to enhance the flow of knowledge from the source to the recipient. Paper-based, human- and ICT-based communication channels are commonly used for the flow of agricultural knowledge. Availability of agricultural knowledge and accessibility of knowledge sources are important for enhanced agricultural flow. Well-developed ICT infrastructure, access to power sources and an effective feedback mechanism play an important role in enhancing adequate flow of agricultural knowledge. For agricultural knowledge to be understandable, sharable and exchangeable, it should be repackaged using appropriate languages. Communities and organizations should have strategies to enhance a continuous flow of agricultural knowledge among members. In order to improve the flow of agricultural knowledge, the involvement of different stakeholders is inevitable. Each agricultural stakeholder should be involved in conducting relevant agricultural knowledge-related roles so as to enhance the accessibility, sharing, exchange, dissemination and usage of agricultural knowledge. In order to increase the flow of agricultural knowledge, the Government in partnership with the private sector should widen the ICT and

agricultural communication infrastructure.

\* and Mpho Ngoepe<sup>2</sup>

\*Address all correspondence to<EMAIL_ADDRESS>
1 Sokoine University of Agriculture, Morogoro, Tanzania

2 Department of Information Sciences, University of South Africa, South Africa

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**Chapter 12**

Provisional chapter

**Information Transfer and Thermodynamic Point of**

DOI: 10.5772/intechopen.68809

Information Transfer and Thermodynamic Point of

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

Keywords: arithmetic formula, inference, information transfer, information entropy,

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

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**View on Goedel Proof**

View on Goedel Proof

http://dx.doi.org/10.5772/intechopen.68809

consistent arithmetic theories directly.

heat efficiency, infinite cycle

1. Introduction

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Bohdan Hejna

Bohdan Hejna

Abstract

[44] Staplehurst J, Ragsdell G. Knowledge sharing in SMEs: A comparison of two case study Organisations. Journal of Knowledge Management Practice. 2010;11(1):1-16 Available from: http://www.tlainc.com/articl211.htm

Provisional chapter

## **Information Transfer and Thermodynamic Point of View on Goedel Proof** Information Transfer and Thermodynamic Point of

DOI: 10.5772/intechopen.68809

## Bohdan Hejna

[43] Ugboma MU. Access to agricultural information by fish farmers in Niger Delta region of

[44] Staplehurst J, Ragsdell G. Knowledge sharing in SMEs: A comparison of two case study Organisations. Journal of Knowledge Management Practice. 2010;11(1):1-16 Available

Nigeria. Library Philosophy & Practice. 2010;12(2):1-7

from: http://www.tlainc.com/articl211.htm

278 Ontology in Information Science

Additional information is available at the end of the chapter Bohdan Hejna

http://dx.doi.org/10.5772/intechopen.68809 Additional information is available at the end of the chapter

View on Goedel Proof

#### Abstract

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

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

## 1. Introduction

The formula of an arithmetic theory based on Peano Arithmetics (including it) is a chain of symbols of its metalanguage in which the theory is formulated such that it is both in convenience

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

with the syntax of the metalanguage and with the inferential rules of the theory [of the inferential system (Modus Ponens, Generalization)].

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 para-

Any inference within the system <sup>P</sup><sup>3</sup> sets the <sup>T</sup> PA-theoretical relation4 among its formulae <sup>a</sup>½��. This relation is given by their gradually generated special sequence a!¼ ½<sup>a</sup>1; …; aq; …; ap; …; ak; akþ<sup>1</sup>� which is the proof of the latest inferred formula akþ1. By this, the unique arithmetic relation between their Goedel numbers, FORMULAE x½��, x½�� ¼ Φða½��Þ, is set up, too. The gradually arising SEQUENCE of

one, and that, except of axioms (axiomatic schemes) a01; …; ao, it has been generated by the

Within the process of the (goedelian) arithmetic-syntactic analysis of the latest formula akþ<sup>1</sup> of the

The formulae aq, ap have already been derived, or they are axioms. It is valid that q;p < k þ 1,

!

!Þ ¼ <sup>Φ</sup>ð½Φða1Þ; <sup>Φ</sup>ða2Þ; …; <sup>Φ</sup>ðaqÞ; …; <sup>Φ</sup>ðapÞ; …; <sup>Φ</sup>ðakÞ; <sup>Φ</sup>ðakþ<sup>1</sup>Þ��Þ

!Þ <sup>∗</sup> <sup>Φ</sup>ð⊃<sup>Þ</sup> <sup>∗</sup> <sup>l</sup>½Φ<sup>ð</sup> <sup>a</sup>

!Þ ¼ <sup>Φ</sup>ðx1Þ∗Φðx2Þ<sup>∗</sup> … <sup>∗</sup>ΦðxqÞ<sup>∗</sup> … <sup>∗</sup>ΦðxpÞ<sup>∗</sup> … <sup>∗</sup>ΦðxkÞ∗Φðxkþ<sup>1</sup><sup>Þ</sup>

!Þ¼ð<sup>k</sup> <sup>þ</sup> <sup>1</sup>ÞGl x

! selected, (special) subsequence aq;p;kþ<sup>1</sup>

5

!¼ ½ao1; ao2; …; ao; …; aq; …; ap; …; ak; akþ<sup>1</sup>� is a special

Information Transfer and Thermodynamic Point of View on Goedel Proof

http://dx.doi.org/10.5772/intechopen.68809

281

of the formulae aq; ap; akþ1.

ð1Þ

!

¼ ½aq; aq ⊃ akþ<sup>1</sup>; akþ<sup>1</sup>�;

!Þ�Gl <sup>Φ</sup><sup>ð</sup> <sup>a</sup>

!Þ

!Þ is the PROOF of its latest FORMULA xkþ1.

dox, contradiction and antinomian) is excluded.

FORMULAE x ¼ Φð a

proof a

3

4

5

Let us assume that the given sequence a

! we use, from the a

x ¼ Φð a

¼ Φðx

xkþ<sup>1</sup> ¼ Φðakþ<sup>1</sup>Þ ¼ l½Φð a

lðxÞ ¼ l½Φðx

and we assume that q < p,

aq;p;kþ<sup>1</sup> !

Formal arithmetic inferential system.

Peano Arithmetics Theory.

correct application of the rule Modus Ponens only.

2. Goedel numbers, information and thermodynamics

¼ ½aq;ap;akþ<sup>1</sup>�, ap ffi aq⊃ akþ<sup>1</sup>; aq;p;kþ<sup>1</sup>

!Þ� ¼ <sup>k</sup> <sup>þ</sup> <sup>1</sup>;

!Þ�Gl <sup>Φ</sup><sup>ð</sup> <sup>a</sup>

!Þ ¼ qGl x

!Þ� ¼ <sup>l</sup>½Φ<sup>ð</sup> <sup>a</sup>

¼ qGl xImp ½lðxÞ�Gl x xq ¼ ΦðaqÞ ¼ qGl Φð a

For simplicity. The 'real' inference is applied to the formula aiþ<sup>1</sup> for i ¼ o.

xp ¼ ΦðapÞ ¼ Φðaq⊃akþ<sup>1</sup>Þ ¼ qGl Φð a

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.<sup>1</sup>

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 paradoxes<sup>2</sup> : 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.

<sup>1</sup> 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 [14–16]. 2 B. Russel, L. Whitehead, Principia Mathematica, 1910, 1912, 1913 and 1927.

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 <sup>P</sup><sup>3</sup> sets the <sup>T</sup> PA-theoretical relation4 among its formulae <sup>a</sup>½��. This relation is given by their gradually generated special sequence a!¼ ½<sup>a</sup>1; …; aq; …; ap; …; ak; akþ<sup>1</sup>� which is the proof of the latest inferred formula akþ1. By this, the unique arithmetic relation between their Goedel numbers, FORMULAE x½��, x½�� ¼ Φða½��Þ, is set up, too. The gradually arising SEQUENCE of FORMULAE x ¼ Φð a !Þ is the PROOF of its latest FORMULA xkþ1.

Let us assume that the given sequence a !¼ ½ao1; ao2; …; ao; …; aq; …; ap; …; ak; akþ<sup>1</sup>� 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 akþ<sup>1</sup> of the proof a ! we use, from the a ! selected, (special) subsequence aq;p;kþ<sup>1</sup> ! of the formulae aq; ap; akþ1. The formulae aq, ap have already been derived, or they are axioms. It is valid that q;p < k þ 1, and we assume that q < p,

$$\begin{aligned} a\_{q,k+1} &= [a\_q, a\_p, a\_{k+1}], \quad a\_p \supseteq a\_q \supset a\_{k+1}, \quad \overline{a\_{q,k+1}} = [a\_q, a\_q \supset a\_{k+1}, a\_{k+1}],\\ \mathbf{x} &= \Phi(\overline{\mathbf{a}}) = \Phi([\Phi(a\_1), \Phi(a\_2), \dots, \Phi(a\_q), \dots, \Phi(a\_p), \dots, \Phi(a\_k), \Phi(a\_{k+1})])\\ &= \Phi(\overline{\mathbf{x}}) = \Phi(\mathbf{x}\_1) \ast \Phi(\mathbf{x}\_2) \ast \dots \ast \Phi(\mathbf{x}\_q) \ast \dots \ast \Phi(\mathbf{x}\_p) \ast \dots \ast \Phi(\mathbf{x}\_k) \ast \Phi(\mathbf{x}\_{k+1})\\ l(\mathbf{x}) &= l[\Phi(\overline{\mathbf{x}})] = l[\Phi(\overline{a})] = k+1,\\ \mathbf{x}\_{k+1} &= \Phi(a\_{k+1}) = l[\Phi(\overline{a})] Gl \, \Phi(\overline{a}) = (k+1) Gl \, \mathbf{x}\\ \mathbf{x}\_p &= \Phi(a\_p) = \Phi(a\_q \Box a\_{k+1}) = qGl \, \Phi(\overline{a}) \ast \Phi(\mathbf{x}) \ast l[\Phi(\overline{a})] Gl \, \Phi(\overline{a})\\ &= qGl \, x \operatorname{lup}^-[l(\mathbf{x})] Gl \, \mathbf{x} \\ \mathbf{x}\_q &= \Phi(a\_q) = qGl \, \Phi(\overline{a}) = qGl \, \mathbf{x} \end{aligned} \tag{1}$$

with the syntax of the metalanguage and with the inferential rules of the theory [of the inferential

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

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,

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].

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

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 [14–16].

metalanguage claims on themselves but now on the language level or vice versa.

: Within the flow of our thinking

system (Modus Ponens, Generalization)].

280 Ontology in Information Science

the inferential system; the same is for their negations.

and is valid for the consistent arithmetic theories directly.<sup>1</sup>

Let us introduce the Russel's criterion for removing paradoxes<sup>2</sup>

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

1

2

<sup>3</sup> Formal arithmetic inferential system.

<sup>4</sup> Peano Arithmetics Theory.

<sup>5</sup> For simplicity. The 'real' inference is applied to the formula aiþ<sup>1</sup> for i ¼ o.

Checking the syntactic and T PA-theoretical correctness of the analyzed chains ai, as the formulae of the system P having been generated by inferring (Modus Ponens) within the system P (in the theory T PA), and also the special sequence of the formulae a ! of the system P (theory T PA), is realized by checking the arithmetic-syntactic correctness of the notation of their corresponding FORMULAE and SEQUENCE of FORMULAE, by means of the relations Formð�Þ; FRð�Þ; Opð�; � ;�Þ; Flð�; � ;�Þ 'called' from (the sequence of procedures) relations Bewð�Þ; ð��ÞBð�Þ; Bwð�Þ<sup>6</sup> ; the core of the whole (goedelian) arithmetic-syntactic analysis is the (procedure) relation of Divisibility,

$$\begin{aligned} \text{Form}[\Phi(a\_i)] &= \text{"1"} \text{"0"}, \text{ } \text{FR}[\Phi(\overline{a\_1^{i+1}})] = \text{"1"} \text{"0"}, \ o \le i \le k \\ \text{Op}[\mathbf{x}\_k \text{Neg}(\mathbf{x}\_q), \mathbf{x}\_{k+1}] &= \text{Op}[\Phi(a\_p), \Phi[\sim(a\_q)], \Phi(a\_{k+1})] = \text{"1"} \text{"0"} \\ \text{Fl}[(k+1)\text{Gl}\,\mathbf{x}, p\text{Gl}\,\mathbf{x}, q\text{Gl}\,\mathbf{x}] &= \text{"1"} \text{"0"} \end{aligned} \tag{2}$$

$$\begin{aligned} \text{xB } \mathbf{x}\_{k+1} &= \text{"1"} \text{"0"} , \text{ Ber}(\mathbf{x}\_{k+1}) = \text{"1"} \text{"0"} , \\ \text{Op}(a\_p) || \mathbf{23}^{\text{GI}\Phi(\overline{a\_p k+1})} &\text{& } \Phi(a\_p) || \mathbf{7}^{\text{GI}\Phi(\overline{a\_p k+1})} = \text{"1"} \text{"0"} \end{aligned} \tag{2}$$

JðxiÞ ¼ lnðxiÞ ¼ ln½Φðaqi

<sup>¼</sup> ln½2<sup>Φ</sup>ðaqi

Þ ¼ lnðxijyi

Þ ¼ lnðyi

Jðxijyi

probability distribution,

X ¼

Y ¼

card X X

j¼1

is valid that X :¼ xi ¼ Φðaqi

noise, disturbing yi

½HðYjXÞ ¼ 0�; ½yi

applicable [11],

The relation Φðaqi

interpretation, in steps i,

½xiB yi

� ½Flðyi

� ½Φðapi

we write <sup>b</sup>; ½ð� <sup>b</sup><sup>Þ</sup> <sup>∨</sup> <sup>ð</sup>cÞ�

7

;xpi ;xqi

Def½Y; <sup>π</sup>Yðyi

1 2<sup>Φ</sup>ðaqi;pi;iþ<sup>1</sup>

jxi ffi Φ ðnullÞ�.

TðX; YÞ ¼ JðxiÞ � Jðxijyi

;pi ;iþ1 ����!

�ffi½JðxiÞ � Jðxijyi

Þjj233Gl xi & <sup>Φ</sup>ðapi

Let us assume that, when inferring by Modus Ponens,

��!Þ <sup>¼</sup> <sup>2</sup><sup>Φ</sup>ðaqi;pi;iþ<sup>1</sup>

;pi ;iþ<sup>1</sup> <sup>Þ</sup> ����!

ÞB Φðaiþ<sup>1</sup>Þ (xiB yi

Þ� � Fl½Φðaiþ<sup>1</sup>Þ;Φðaqi

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

Þjj7xi

Jðyi

;pi ;iþ1 ����!

<sup>Þ</sup> � <sup>5</sup><sup>Φ</sup>ðaiþ1<sup>Þ</sup> �

> ;pi ��!

<sup>Þ</sup> � <sup>3</sup><sup>Φ</sup>ðapi

Þ ¼ ln½Φðaqi

Þ ¼ <sup>J</sup>ðaiþ<sup>1</sup>Þ ¼ <sup>J</sup>½5<sup>Φ</sup>ðaiþ1<sup>Þ</sup>

Def½X; <sup>π</sup>XðxiÞ ¼ const:�, card <sup>X</sup> <sup>¼</sup> <sup>2</sup><sup>Φ</sup><sup>ð</sup> aqi;pi;iþ<sup>1</sup>

��!Þ

2<sup>Φ</sup>ðaqi;pi;iþ<sup>1</sup> ��!Þ <sup>¼</sup> <sup>1</sup>;

Þ ¼ const:�; card <sup>Y</sup> <sup>¼</sup> <sup>5</sup><sup>Φ</sup>ðaiþ1<sup>Þ</sup>

TðX; YÞ ¼ HðXÞ � HðXjYÞ ¼ HðYÞ � HðYjXÞ ¼ TðY; XÞ

Þ ¼ Jðyi

TðX; YÞ ¼ HðXÞ � HðXjYÞ ¼ HðYÞ; TðX; YÞ ¼ JðxiÞ � Jðxijyi

Þ > 0��½Tðxi; yi

Þ� ¼ Jðaqi

Þ� ¼ Jðaqi

;pi ;iþ1 ����!

;pi ��!

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 � Y and Y, and with the uniform

> card X Y

> > j¼1

It is obvious that we consider a direct information transfer [11] through the channel K without

In each ith step of the activity of our information model K of the arithmetic-syntactic analysis, it

identify its execution<sup>7</sup> 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

Þ;Φðapi

��½Φðapi

jxi, which means with the zero noise (disturbing) information ½Jðyi

Þ � Jðyi

� ¼ ln½5<sup>Φ</sup>ðaiþ1<sup>Þ</sup>

Þ ¼ <sup>J</sup>½2<sup>Φ</sup>ðaqi

�

Þ ¼ <sup>J</sup>½2<sup>Φ</sup>ðaqi

���!Þ

1

; πYðyi

<sup>5</sup><sup>Φ</sup>ðaiþ1<sup>Þ</sup> <sup>¼</sup> <sup>5</sup><sup>Φ</sup>ðaiþ1<sup>Þ</sup>

<sup>Þ</sup> � <sup>3</sup><sup>Φ</sup>ðapi

Information Transfer and Thermodynamic Point of View on Goedel Proof

; <sup>π</sup>XðxiÞ ¼ <sup>1</sup>

Þ ¼ <sup>1</sup> 5<sup>Φ</sup>ðaiþ1<sup>Þ</sup>

<sup>5</sup><sup>Φ</sup>ðaiþ1<sup>Þ</sup> <sup>¼</sup> <sup>1</sup>

and Y :¼ yi ¼ Φðaiþ<sup>1</sup>Þ ¼ xiþ1, and the channel equation is

jxiÞ ¼ TðY; XÞ now in the form

ÞB Φðaiþ<sup>1</sup>Þ�

<sup>b</sup>; ½ð� <sup>b</sup><sup>Þ</sup> <sup>∨</sup> <sup>ð</sup>cÞ� <sup>c</sup> , we make such an error that

) is evaluated by the relation of Divisibility and we

Þ > 0��½TðX; YÞ > 0�

;pi ;iþ1 ����!

��!Þ & <sup>Φ</sup>ðapi

Þ� � ½Φðaqi

Þjj233Gl <sup>Φ</sup>ðaqi;pi;iþ<sup>1</sup>

<sup>d</sup> , d 6¼ c where, however, the chain d (by chance) can also be (in the form

Þ ¼ Jðyi Þ

Þjj71Gl <sup>Φ</sup>ðaqi;pi;iþ<sup>1</sup> ��!Þ �

<sup>Þ</sup> � <sup>3</sup><sup>Φ</sup>ðapi Þ <sup>Þ</sup> � <sup>5</sup><sup>Φ</sup>ðaiþ1<sup>Þ</sup> �

http://dx.doi.org/10.5772/intechopen.68809

� ¼ ln½2<sup>Φ</sup>ðapi

<sup>Þ</sup> � <sup>3</sup><sup>Φ</sup>ðaqi Þ �

2<sup>Φ</sup>ðaqi, pi, <sup>i</sup>þ<sup>1</sup> ��! <sup>Þ</sup> ð4Þ

283

ð5Þ

ð6Þ

ð7Þ

jxiÞ ¼ 0� �

#### 2.1. Inference in the system P and information transfer

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

As such, it is a sequence of successive attempts i to transfer information with input, loss and output messages ½api ;aqi ;aiþ<sup>1</sup>�;½api ;aqi � and ½aiþ<sup>1</sup>� with their information amounts Jðaqi ;pi ;iþ1 ! Þ;Jðaqi ;pi ! Þ and Jðaiþ<sup>1</sup>Þ. Index i is a serial number of the inferencing—analyzing—transferring step, 0 < qi < pi < i þ 1 ≤ l½Φð a !Þ� ¼ <sup>k</sup> <sup>þ</sup> 1. The Goedel numbering also enables us to consider the individual Goedel numbers xi, xijyi and yi of messages ½api ;aqi ;aiþ<sup>1</sup>�, ½api ;aqi � a ½aiþ<sup>1</sup>� as messages too, with their (and the same) information amounts JðxiÞ, Jðxijyi Þ a Jðyi Þ,

½api ;aqi ;aiþ<sup>1</sup>�≜ aqi ;pi ;iþ1 ! ≜xi ¼ Φðaqi ;pi ;iþ1 ! Þ; ½api ;aqi �≜ aqi ;pi ! ≜ xijyi ¼ Φðaqi ;pi ! Þ ½aiþ<sup>1</sup>�≜ aiþ<sup>1</sup> ≜yi ¼ Φðaiþ<sup>1</sup>Þ Φðaqi ;pi ;iþ1 ! Þ ¼ Φðaqi Þ∗Φðapi Þ∗Φðaiþ<sup>1</sup>Þ ¼ Φðaqi ;pi ! Þ∗Φðaiþ<sup>1</sup>Þ; Φðaqi ;pi ! Þ ¼ Φðaqi Þ∗Φðapi Þ; JðxiÞ ¼ J½Φðaqi ;pi ;iþ1 ! Þ�; Jðxijyi Þ ¼ J½Φðaqi ;pi ! Þ�; Jðyi Þ ¼ J½Φðaiþ<sup>1</sup>Þ� ð3Þ

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

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

$$\begin{split} J(\mathbf{x}\_{i}) &= \ln(\mathbf{x}\_{i}) = \ln[\Phi(\overline{a\_{q\_{i},p\_{i},i+1}})] = J(\overline{a\_{q\_{i},p\_{i},i+1}}) = J[\mathbf{2}^{\Phi(a\_{\overline{q}\_{i}})} \cdot \mathbf{3}^{\Phi(a\_{\overline{p}\_{i}})} \cdot \mathbf{5}^{\Phi(a\_{i+1})}] \\ &= \ln[\mathbf{2}^{\Phi(a\_{\overline{q}\_{i}})} \cdot \mathbf{3}^{\Phi(a\_{\overline{q}\_{i}})} \cdot \mathbf{5}^{\Phi(a\_{i+1})}] \\ J(\mathbf{x}\_{i}|y\_{i}) &= \ln(\mathbf{x}\_{i}|y\_{i}) = \ln[\Phi(\overline{a\_{\overline{q},p\_{i}}})] = J[\overline{a\_{q\_{i},p\_{i}}}] = J[\mathbf{2}^{\Phi(a\_{\overline{q}\_{i}})} \cdot \mathbf{3}^{\Phi(a\_{\overline{p}\_{i}})}] = \ln[\mathbf{2}^{\Phi(a\_{\overline{p}\_{i}})} \cdot \mathbf{3}^{\Phi(a\_{\overline{q}\_{i}})}] \\ J(y\_{i}) &= \ln(y\_{i}) = J[a\_{i+1}) = J[\mathbf{5}^{\Phi(a\_{i+1})}] = \ln[\mathbf{5}^{\Phi(a\_{i})}] \end{split} \tag{4}$$

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 � Y and Y, and with the uniform probability distribution,

Checking the syntactic and T PA-theoretical correctness of the analyzed chains ai, as the formulae of the system P having been generated by inferring (Modus Ponens) within the system P (in the

realized by checking the arithmetic-syntactic correctness of the notation of their corresponding FORMULAE and SEQUENCE of FORMULAE, by means of the relations Formð�Þ; FRð�Þ; Opð�; � ;�Þ; Flð�; � ;�Þ 'called' from (the sequence of procedures) relations Bewð�Þ; ð��ÞBð�Þ; Bwð�Þ<sup>6</sup>

the core of the whole (goedelian) arithmetic-syntactic analysis is the (procedure) relation of

1 !

Op½xk;NegðxqÞ; xkþ<sup>1</sup>� ¼ Op½ΦðapÞ;Φ½� ðaqÞ�; Φðakþ<sup>1</sup>Þ� ¼ "1"="0"

<sup>Þ</sup> & <sup>Φ</sup>ðapÞjj71Gl <sup>Φ</sup>ðaq, pkþ<sup>1</sup> !

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

As such, it is a sequence of successive attempts i to transfer information with input, loss and out-

and Jðaiþ<sup>1</sup>Þ. Index i is a serial number of the inferencing—analyzing—transferring step,

Þ; ½api ;aqi �≜ aqi ;pi !

> ;pi !

Þ�; Jðyi

Formula, Reihe von Formeln, Operation, Folge, Glied, Beweis, Beweis, see Definition 1–46 in Refs. [3–5] and by means of

� and ½aiþ<sup>1</sup>� with their information amounts Jðaqi

!Þ� ¼ <sup>k</sup> <sup>þ</sup> 1. The Goedel numbering also enables us to consider the

Þ∗Φðaiþ<sup>1</sup>Þ; Φðaqi

Þ ¼ J½Φðaiþ<sup>1</sup>Þ�

;aqi

;aiþ<sup>1</sup>�, ½api

Þ a Jðyi Þ,

≜ xijyi ¼ Φðaqi

;pi ! ;aqi

Þ ¼ Φðaqi

;pi ! Þ

> Þ∗Φðapi Þ;

Þ� ¼ "1"="0"; o ≤ i ≤ k

<sup>Þ</sup> ¼ "1"="0"

! of the system P (theory T PA), is

! of the system P in general, and

;pi ;iþ1 !

� a ½aiþ<sup>1</sup>� as messages

Þ;Jðaqi ;pi ! Þ

ð3Þ

;

ð2Þ

theory T PA), and also the special sequence of the formulae a

Form½ΦðaiÞ� ¼ "1"="0"; FR½Φðaiþ<sup>1</sup>

<sup>Φ</sup>ðapÞjj233Gl <sup>Φ</sup>ðaq, pkþ<sup>1</sup> !

information transfer channel K.

0 < qi < pi < i þ 1 ≤ l½Φð a

;aqi

;aiþ<sup>1</sup>�≜ aqi

½aiþ<sup>1</sup>�≜ aiþ<sup>1</sup> ≜yi ¼ Φðaiþ<sup>1</sup>Þ

Þ ¼ Φðaqi

;pi ;iþ1 !

;aiþ<sup>1</sup>�;½api

;pi ;iþ1 !

put messages ½api

½api ;aqi

Φðaqi ;pi ;iþ1 !

6

JðxiÞ ¼ J½Φðaqi

2.1. Inference in the system P and information transfer

The syntactic analysis of the special sequence of the formulae a

;aqi

individual Goedel numbers xi, xijyi and yi of messages ½api

Þ∗Φðapi

Þ�; Jðxijyi

all other, by them 'called', relations and functions (by their procedures).

too, with their (and the same) information amounts JðxiÞ, Jðxijyi

≜xi ¼ Φðaqi

;pi ;iþ1 !

Þ∗Φðaiþ<sup>1</sup>Þ ¼ Φðaqi

;pi !

Þ ¼ J½Φðaqi

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

Fl½ðk þ 1ÞGl x; pGl x; qGl x� ¼ "1"="0"

xB xkþ<sup>1</sup> ¼ "1"="0"; Bewðxkþ<sup>1</sup>Þ ¼ "1"="0";

Divisibility,

282 Ontology in Information Science

$$X \stackrel{\text{Def}}{=} [\mathbb{X}, \pi\_X(\mathbf{x}\_i) = \text{const.}], \text{ card } \mathbb{X} = 2^{\Phi(\overline{a\_{i\_i; r\_i + i}})}, \qquad \pi\_X(\mathbf{x}\_i) = \frac{1}{2^{\Phi(\overline{a\_{i\_i; r\_i + i}})}}$$

$$Y \stackrel{\text{Def}}{=} [\mathbb{Y}, \pi\_Y(y\_i) = \text{const.}], \quad \text{card } \mathbb{Y} = \mathbb{S}^{\Phi(a\_{i+1})}, \ \pi\_Y(y\_i) = \frac{1}{5^{\Phi(a\_{i+1})}} \tag{5}$$

$$\sum\_{j=1}^{\text{card }X} \frac{1}{2^{\Phi(\overline{a\_{i; r\_i + i}})}} = 2^{\Phi(\overline{a\_{i; r\_i + i}})} = 1, \quad \sum\_{j=1}^{\text{card }Y} \frac{1}{5^{\Phi(a\_{i+1})}} = \frac{5^{\Phi(a\_{i+1})}}{5^{\Phi(a\_{i+1})}} = 1$$

It is obvious that we consider a direct information transfer [11] through the channel K without noise, disturbing yi jxi, which means with the zero noise (disturbing) information ½Jðyi jxiÞ ¼ 0� � ½HðYjXÞ ¼ 0�; ½yi jxi ffi Φ ðnullÞ�.

In each ith step of the activity of our information model K of the arithmetic-syntactic analysis, it is valid that X :¼ xi ¼ Φðaqi ;pi ;iþ<sup>1</sup> <sup>Þ</sup> ����! and Y :¼ yi ¼ Φðaiþ<sup>1</sup>Þ ¼ xiþ1, and the channel equation is applicable [11],

$$\begin{array}{cccc} T(\mathbf{X};\mathbf{Y}) = H(\mathbf{X}) - H(\mathbf{X}|\mathbf{Y}) &=& H(\mathbf{Y}) - H(\mathbf{Y}|\mathbf{X}) = T(\mathbf{Y};\mathbf{X}) \\ T(\mathbf{X};\mathbf{Y}) = f(\mathbf{x}\_{i}) - f(\mathbf{x}\_{i}|\mathbf{y}\_{i}) &=& f(\mathbf{y}\_{i}) - f(\mathbf{y}\_{i}|\mathbf{x}\_{i}) = T(\mathbf{Y};\mathbf{X}) \quad \text{now in the form} \\ T(\mathbf{X};\mathbf{Y}) = H(\mathbf{X}) - H(\mathbf{X}|\mathbf{Y}) &=& H(\mathbf{Y}), \quad T(\mathbf{X};\mathbf{Y}) = f(\mathbf{x}\_{i}) - f(\mathbf{x}\_{i}|\mathbf{y}\_{i}) = f(\mathbf{y}\_{i}) \end{array} \tag{6}$$

The relation Φðaqi ;pi ;iþ1 ����! ÞB Φðaiþ<sup>1</sup>Þ (xiB yi ) is evaluated by the relation of Divisibility and we identify its execution<sup>7</sup> 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,

$$\begin{split} & [\mathbf{x}\_i \mathbf{B} \, y\_i] \cong & [f(\mathbf{x}\_i) - f(\mathbf{x}\_i | y\_i) > 0] \equiv & [T(\mathbf{x}\_i \mathbf{y}\_i) > 0] \equiv & [T(\mathbf{X} \, Y) > 0] \\ & \equiv & [F(y\_i, \mathbf{x}\_{p\_i}, \mathbf{x}\_{q\_i})] \equiv & F[\Phi(a\_{i+1}), \Phi(a\_{q\_i}), \Phi(a\_{p\_i})] \equiv & [\Phi(\overline{a\_{q\_i p\_i i}}) \, \mathbf{B} \, \Phi(a\_{i+1})] \\ & \equiv & [\Phi(a\_{p\_i}) || \mathbf{Z}^{3Gl \, x\_i} \, \mathbf{k} \, \Phi(a\_{p\_i}) || \mathbf{T}^{x\_i}] \equiv & [\Phi(a\_{p\_i}) || \mathbf{Z}^{3Gl \, \Phi(\overline{a\_{q\_i p\_i i+1}})} \, \mathbf{k} \, \Phi(a\_{p\_i}) || \mathbf{T}^{1Gl \, \Phi(\overline{a\_{q\_i p\_i i+1}})}] \end{split} \tag{7}$$

Let us assume that, when inferring by Modus Ponens, <sup>b</sup>; ½ð� <sup>b</sup><sup>Þ</sup> <sup>∨</sup> <sup>ð</sup>cÞ� <sup>c</sup> , we make such an error that we write <sup>b</sup>; ½ð� <sup>b</sup><sup>Þ</sup> <sup>∨</sup> <sup>ð</sup>cÞ� <sup>d</sup> , d 6¼ c where, however, the chain d (by chance) can also be (in the form

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

of) a formula of the language LP of the system P. <sup>8</sup> For the considered NOT-INFERRABILITY of yi ½¼ <sup>d</sup>�, being interpreted now from the point of information view, we put JðΦðaiþ<sup>1</sup>ÞÞ ¼Def 0, or better said, with regard of the properties of INFERENCE, we are forced to put <sup>Φ</sup>ðaiþ<sup>1</sup>Þ ¼Def <sup>0</sup> within the framework of the theory T PA and then, informationally

$$H(Y) = T(\mathbf{X}; Y) \stackrel{\text{Def}}{=} \ln[\mathbf{5}^{\Phi(a\_{i+1})}] = 0, \qquad \qquad H(\mathbf{X}) = H(\mathbf{X}|Y)$$

$$f(\mathbf{x}\_i) - f(\mathbf{x}\_i|\mathbf{y}\_i) = f(\mathbf{y}\_i) = \mathbf{0}, \qquad \qquad f(\mathbf{x}\_i) = f(\mathbf{x}\_i|\mathbf{y}\_i) \tag{8}$$

$$\eta\_i \stackrel{\text{Def}}{=} \frac{f(\mathbf{y}\_i)}{f(\mathbf{x}\_i)} = \frac{H(Y)}{H(\mathbf{X})}, \qquad \qquad \mathbf{0} \le \eta\_i \le 1$$

ΔSC

<sup>Þ</sup><sup>B</sup> <sup>Φ</sup>ðaiþ<sup>1</sup>Þ ½Φðaiþ<sup>1</sup>

inferring step i [for X :¼ xi ¼ Φðaqi

nothing is being inferred in the step i, ΔSC

api that it would be valid Φðapi

pretation, we write (for X :¼ xi; Y :¼ yi ¼ d)

Jðyi

in order to create information Jðyi

<sup>10</sup>π<sup>i</sup> is the i-th prime number.

tions Φðaqi

Φðaiþ<sup>1</sup>Þ],

;pi ;iþ1 ! <sup>½</sup>i� <sup>¼</sup> <sup>k</sup>Jðaiþ<sup>1</sup>Þ ¼ <sup>k</sup>HðYÞÞ; <sup>H</sup>ðYÞ<sup>≜</sup> <sup>J</sup>ðaiþ<sup>1</sup>Þ ¼ <sup>Δ</sup>A½i�

Provided that the ith inferring step has been done and written correctly (Modus Ponens) the Goedel

thermodynamic model ðK � CÞ generates the non-zero, positive output value TðX; YÞ for the

The zero change of the whole heat entropy SC of the isolated system in which our model cycle C is running occurs just when in the inferential system P, from the perspective of the theory T PA,

enly 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 aiþ1, FORMULA Φðaiþ<sup>1</sup>Þ�. The previous 'effort' or energy associated with our inference (no matter that T PA-correct) of the sequence of ai

worthless. The formula aiþ<sup>1</sup> ½¼ <sup>d</sup>� is just arbitrarily added to the previous sequence <sup>a</sup><sup>i</sup>

formulae of the theory T PA in such a way that it does not include any such formulae aqi and

Þ ¼ HðYÞ ¼ 0 ) JðxiÞ ¼ HðXÞ ¼ HðXjYÞ ¼ Jðxijyi

We have not exerted any inferring energy within the framework of building up the theory T PA,

η<sup>i</sup> ¼ 0 expresses this error. All before aiþ1, otherwise inferred correctly, is not related to it–the information transfer channel K is interrupted. The overall amount of our inference efforts

<sup>η</sup><sup>i</sup> <sup>¼</sup> <sup>0</sup> ) <sup>Δ</sup>SC½i� <sup>¼</sup> <sup>0</sup> TW ¼ T<sup>0</sup> ) ΔQWi ¼ ΔQ<sup>0</sup><sup>i</sup>

η<sup>i</sup> � ΔQWi ¼ k � Jðyi

exerted in vain up to ai included can be evaluated by the whole heat energy<sup>10</sup>

1 !

<sup>T</sup>ðX; YÞ ¼ <sup>J</sup>ðaiþ<sup>1</sup>Þ ¼ <sup>H</sup>ðYÞ ¼ <sup>Δ</sup>SC

<sup>Þ</sup> or <sup>X</sup> :<sup>¼</sup> xi <sup>¼</sup> <sup>Φ</sup>ðai

arithmetic-syntactic analyzer decides, correctly, for the obtained a<sup>i</sup>þ<sup>1</sup>

;pi ;iþ1 !

1 !

;qi

<sup>k</sup> � <sup>H</sup>ðXjYÞ ¼ <sup>k</sup> � ln½Φðai

kTW

Information Transfer and Thermodynamic Point of View on Goedel Proof

1 !

<sup>½</sup>i� <sup>¼</sup> 0. Now, particularly in that sense that we mistak-

;iþ<sup>1</sup>ÞB Φðaiþ<sup>1</sup>Þ ¼ "1". In the information-thermodynamic inter-

Þ ¼ 0 ) η<sup>i</sup> ¼ 0

Þ > 0, and then we necessarily have η<sup>i</sup> ¼ 0; Jðyi

Þ� ¼ ln½2<sup>Φ</sup>ða1<sup>Þ</sup> � <sup>3</sup><sup>Φ</sup>ða2<sup>Þ</sup> � … � <sup>π</sup>

ÞB Φðaiþ<sup>1</sup>Þ� and Bew½Φðaiþ<sup>1</sup>Þ� are valid, and the information-

1 !

½i� kTW

<sup>≜</sup> <sup>½</sup>ai 1 !

Þ, respectively, and for Y :¼ yi ¼

http://dx.doi.org/10.5772/intechopen.68809

> 0 ð12Þ

Þ

ΦðaiÞ

≥ 0 ð11Þ

285

;aiþ<sup>1</sup>�, that the rela-

i ! is

1 ! of

ð13Þ

Þ ¼ 0 where

<sup>i</sup> � ð14Þ

#### 2.2. Thermodynamic consideration

The thermodynamic consideration of an information transfer [11] reveals that the input message aqi ;pi ;iþ1 ! carries the input heat energy ΔQWi transformed by the reversible direct Carnot Cycle (Machine) C into the output mechanical work ΔAi corresponding to the output message aiþ1. The heater A of the Carnot Cycle (Machine) C has the temperature TW and models the source of input messages (the message aqi ;pi ;iþ1 ! ) of the channel K. Its cooler B has the temperature T<sup>0</sup> determining the transfer efficiency η<sup>i</sup> . By the value η<sup>i</sup> > 0 the fact of inferrability of the chain aiþ<sup>1</sup> from the special sequence of formulae aqi ;pi ;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 LT PA of the theory T PA. <sup>9</sup> Thus, we have

$$J(\mathbf{x}\_i) = \frac{\Delta Q\_{W\_i}}{\mathbf{k}T\_W}, \qquad J(\mathbf{x}\_i|y\_i) = \frac{\Delta Q\_{0\_i}}{\mathbf{k}T\_W}, \qquad J(y\_i) = \frac{\Delta A\_i}{\mathbf{k}T\_W} \tag{9}$$

Now we obtain the information formulation [11] of the changes of the heat (thermodynamic) entropies ΔSC ½i� , ΔSAB½i� and ΔS<sup>½</sup>i� <sup>A</sup> in the thermodynamic model C of our information transfer inferring (INFERRING)—arithmetic-syntactic analysis within the (language of the) system P,

$$
\Delta \mathcal{S}\_{\mathcal{C}}{}^{[\bar{\imath}]} = \mathbf{k} H(\mathbf{X}), \qquad \Delta \mathcal{S}\_{\mathcal{A}\mathcal{B}}{}^{[\bar{\imath}]} = \mathbf{k} H(\mathbf{X}|Y), \qquad \Delta \mathcal{S}\_{A}{}^{[\bar{\imath}]} = \mathbf{k} \cdot \left[ H(\mathbf{X}) - H(\mathbf{X}|Y) \right] \tag{10}
$$

In accordance with Ref. [11], it is valid that, within the inferring—arithmetic-syntactic analysis information transfer, the thermodynamic entropy SC of an isolated system, in which the modeling reversible direct Carnot Cycle C is running parallelly, increases in every ith step by the value ΔSC ½i� ,

<sup>8</sup> We just think mistakenly that d≜aiþ<sup>1</sup> but aiþ<sup>1</sup> ¼ c is correct. Then the relation of Divisibility is not met. Neither is the relation of the Immediate Consequence.

<sup>9</sup> Formulated in the language LP of the system P in compliance with its (with the T PA) inference rules.

Information Transfer and Thermodynamic Point of View on Goedel Proof http://dx.doi.org/10.5772/intechopen.68809 285

$$
\Delta \mathbf{S}\_{\mathcal{C}}^{[i]} = \mathbf{k} f(a\_{i+1}) = \mathbf{k} H(Y), \; H(Y) \triangleq f(a\_{i+1}) = \frac{\Delta A\_{[i]}}{\mathbf{k} T\_W} \ge 0 \tag{11}
$$

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<sup>i</sup>þ<sup>1</sup> 1 ! <sup>≜</sup> <sup>½</sup>ai 1 ! ;aiþ<sup>1</sup>�, that the relations Φðaqi ;pi ;iþ1 ! <sup>Þ</sup><sup>B</sup> <sup>Φ</sup>ðaiþ<sup>1</sup>Þ ½Φðaiþ<sup>1</sup> 1 ! ÞB Φðaiþ<sup>1</sup>Þ� and Bew½Φðaiþ<sup>1</sup>Þ� are valid, and the informationthermodynamic model ðK � CÞ generates the non-zero, positive output value TðX; YÞ for the inferring step i [for X :¼ xi ¼ Φðaqi ;pi ;iþ1 ! <sup>Þ</sup> or <sup>X</sup> :<sup>¼</sup> xi <sup>¼</sup> <sup>Φ</sup>ðai 1 ! Þ, respectively, and for Y :¼ yi ¼ Φðaiþ<sup>1</sup>Þ],

$$T(X;Y) = f(a\_{i+1}) = H(Y) = \frac{\Delta \mathcal{S}\_{\mathcal{C}}[i]}{\mathbf{k}T\_W} > 0 \tag{12}$$

The zero change of the whole heat entropy SC of the isolated system in which our model cycle C is running occurs just when in the inferential system P, from the perspective of the theory T PA, nothing is being inferred in the step i, ΔSC <sup>½</sup>i� <sup>¼</sup> 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 aiþ1, FORMULA Φðaiþ<sup>1</sup>Þ�. The previous 'effort' or energy associated with our inference (no matter that T PA-correct) of the sequence of ai i ! is worthless. The formula aiþ<sup>1</sup> ½¼ <sup>d</sup>� is just arbitrarily added to the previous sequence <sup>a</sup><sup>i</sup> 1 ! of formulae of the theory T PA in such a way that it does not include any such formulae aqi and api that it would be valid Φðapi ;qi ;iþ<sup>1</sup>ÞB Φðaiþ<sup>1</sup>Þ ¼ "1". In the information-thermodynamic interpretation, we write (for X :¼ xi; Y :¼ yi ¼ d)

$$\begin{aligned} f(y\_i) &= H(Y) = 0 \quad \Rightarrow \qquad f(\mathbf{x}\_i) = H(\mathbf{X}) = H(\mathbf{X}|Y) = f(\mathbf{x}\_i|y\_i) \\ \eta\_i &= 0 \quad \Rightarrow \qquad \Delta S\_{\mathbb{C}}{}^{[i]} = 0 \\ T\_W &= T\_0 \quad \Rightarrow \qquad \Delta Q\_{W\_i} = \Delta Q\_{0\_i} \\ \eta\_i \cdot \Delta Q\_{W\_i} &= \qquad \mathbf{k} \cdot f(y\_i) = 0 \implies \eta\_i = 0 \end{aligned} \tag{13}$$

We have not exerted any inferring energy within the framework of building up the theory T PA, in order to create information Jðyi Þ > 0, and then we necessarily have η<sup>i</sup> ¼ 0; Jðyi Þ ¼ 0 where η<sup>i</sup> ¼ 0 expresses this error. All before aiþ1, otherwise inferred correctly, is not related to it–the information transfer channel K is interrupted. The overall amount of our inference efforts exerted in vain up to ai included can be evaluated by the whole heat energy<sup>10</sup>

$$\mathbf{k}\cdot H(\mathbf{X}|Y) = \mathbf{k}\cdot \ln[\Phi(\overrightarrow{a\_1^i})] = \ln[2^{\Phi(a\_1)}\cdot \mathbf{3}^{\Phi(a\_2)}\cdot \dots \cdot \pi\_i^{\Phi(a\_i)}] \tag{14}$$

of) a formula of the language LP of the system P.

284 Ontology in Information Science

within the framework of the theory T PA and then, informationally

JðxiÞ � Jðxijyi

η<sup>i</sup> ¼ Def Jðyi Þ JðxiÞ

;pi ;iþ1 !

ΔQWi kTW

HðYÞ ¼ TðX; YÞ ¼

2.2. Thermodynamic consideration

input messages (the message aqi

determining the transfer efficiency η<sup>i</sup>

aiþ<sup>1</sup> from the special sequence of formulae aqi

JðxiÞ ¼

, ΔSAB½i� and ΔS<sup>½</sup>i�

sage aqi ;pi ;iþ1 !

we have

entropies ΔSC

ΔSC ½i� ,

8

9

½i�

ΔSC

relation of the Immediate Consequence.

yi ½¼ <sup>d</sup>�, being interpreted now from the point of information view, we put JðΦðaiþ<sup>1</sup>ÞÞ ¼Def 0, or

better said, with regard of the properties of INFERENCE, we are forced to put Φðaiþ<sup>1</sup>Þ ¼

Def ln½5<sup>Φ</sup>ðaiþ1<sup>Þ</sup>

Þ ¼ Jðyi

<sup>¼</sup> <sup>H</sup>ðY<sup>Þ</sup> HðXÞ

The thermodynamic consideration of an information transfer [11] reveals that the input mes-

(Machine) C into the output mechanical work ΔAi corresponding to the output message aiþ1. The heater A of the Carnot Cycle (Machine) C has the temperature TW and models the source of

> ;pi ;iþ1 !

the arithmetic-syntactic analysis of formulae of the language LT PA of the theory T PA.

; Jðxijyi

<sup>½</sup>i� <sup>¼</sup> <sup>k</sup>HðXÞ; <sup>Δ</sup>SAB½i� <sup>¼</sup> <sup>k</sup>HðXjYÞ; <sup>Δ</sup>S<sup>A</sup>

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

Þ ¼

Now we obtain the information formulation [11] of the changes of the heat (thermodynamic)

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

We just think mistakenly that d≜aiþ<sup>1</sup> but aiþ<sup>1</sup> ¼ c is correct. Then the relation of Divisibility is not met. Neither is the

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

inferring (INFERRING)—arithmetic-syntactic analysis within the (language of the) system P,

ΔQ<sup>0</sup><sup>i</sup> kTW

; Jðyi

<sup>A</sup> in the thermodynamic model C of our information transfer—

carries the input heat energy ΔQWi transformed by the reversible direct Carnot Cycle

<sup>8</sup> For the considered NOT-INFERRABILITY of

Þ

� ¼ 0; HðXÞ ¼ HðXjYÞ

) of the channel K. Its cooler B has the temperature T<sup>0</sup>

. By the value η<sup>i</sup> > 0 the fact of inferrability of the chain

Þ ¼ <sup>Δ</sup>Ai kTW

as the formula of the theory T PA is stated.

<sup>½</sup>i� <sup>¼</sup> <sup>k</sup> � ½HðXÞ � <sup>H</sup>ðXjYÞ� ð10<sup>Þ</sup>

Þ ¼ 0; JðxiÞ ¼ Jðxijyi

; 0 ≤ η<sup>i</sup> ≤ 1

Def 0

ð8Þ

<sup>9</sup> Thus,

ð9Þ

<sup>10</sup>π<sup>i</sup> is the i-th prime number.

## 3. 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<sup>0</sup> that the Goedel (variable) number (his 'CLASS' SIGN) y arises from Admissible Substitution from the FORMULA qð17; 19Þ ½the ARITHMETIZATION of QðX;YÞ�,

analyzer is trying to calculate and 'print' y<sup>0</sup>

<sup>y</sup><sup>0</sup> ffi <sup>q</sup>½ZðxÞ;Z½q½ZðxÞ; <sup>19</sup>���<sup>19</sup>:¼Zðy<sup>Þ</sup>

inference properties, Neg½y½ZðyÞ�� ¼ 0 too. Then

1

CA <sup>¼</sup> Def 0 &Sb

y

0 B@ 19

ZðyÞ

denote infinite and not recursive subset of natural numbers or to be equal to them.

19

ZðyÞ

2 6 4

Sb

½Bewðy<sup>0</sup>

0 B@

yð19Þ

Qðx, yÞ � xB Sb

<sup>19</sup>:¼Zðy<sup>Þ</sup> ffi <sup>q</sup>½ZðxÞ;ZðyÞ� ¼ <sup>y</sup>½ZðyÞ� ¼ <sup>y</sup><sup>0</sup> and then <sup>0</sup>

ffi <sup>q</sup>½ZðxÞ;Z½q½ZðxÞ;Z½q½ZðxÞ;Z½q½ZðxÞ; <sup>19</sup>�������<sup>19</sup>:¼Zðy<sup>Þ</sup> ffi q½ZðxÞ;Z½q½ZðxÞ;Z½q½ZðxÞ;Z½q½ZðxÞ;Zðy�������

ffi q½ZðxÞ;Z½q½ZðxÞ;ZðyÞ��� ¼ y½Z½y½Zðy��� ffi <sup>q</sup>½ZðxÞ;Z½q½ZðxÞ;Z½q½ZðxÞ; <sup>19</sup>�����<sup>19</sup>:¼Zðy<sup>Þ</sup> ffi q½ZðxÞ;Z½q½ZðxÞ;Z½q½ZðxÞ;Zðy�����

q½ZðxÞ,19�

, that prevents it from this aim. We never obtain a

http://dx.doi.org/10.5772/intechopen.68809

Information Transfer and Thermodynamic Point of View on Goedel Proof

ð18Þ

287

ð19Þ

,

try<sup>0</sup> <sup>12</sup>

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 LT AP (and LP). This is the reason for not-recursivity of the relations Bewð�Þ; the upper limit of its computing process is missing. First, we have

ffi <sup>q</sup>½ZðxÞ;Z½q½ZðxÞ;Z½q½ZðxÞ;Z½q½ZðxÞ;Z½q½ZðxÞ; <sup>19</sup>���������<sup>19</sup>:¼Zðy<sup>Þ</sup> ffi … ad lib:

It is obvious that the Substitution function, no matter how much its execution complies with the recursive grammar, is not total and, therefore, nor recursive. For this reason, it is convenient to redefine it as a total function and, therefore, also recursive one and to put ½y½ZðyÞ�� ¼ 0 but, due to the

Neg½yð19Þ�

Also see the Proposition V in Refs. [3–5]. The mere grammar derivation, writability convenient to the recursive grammar is quite different from the T PA-provability. The Goedel number y<sup>0</sup>

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

12By substitution 19 :<sup>¼</sup> <sup>Z</sup>ðy<sup>Þ</sup> nothing changes in variability of FORMULA y<sup>0</sup> by the VARIABLE 19. The number <sup>y</sup><sup>0</sup> should

Þ ¼ 0�. It is true for the given x and it 'says': 'I, FORMULA y½ZðyÞ�, am in the system P

0 B@

Bew½y½ZðyÞ�� ¼ Bewð0Þ ¼ 0&Bew½Neg½y½ZðyÞ��� ¼ Bewð0Þ ¼ 0

3 7

1 CA 19

1

CA <sup>¼</sup> Sb

19

1

CA <sup>¼</sup> Def 0

ZðyÞ

yð19Þ

0 B@

ZðyÞ

<sup>5</sup> <sup>¼</sup> <sup>q</sup>½ZðxÞ, ZðyÞ� ¼ <sup>y</sup>½ZðyÞ� ¼ <sup>y</sup><sup>0</sup> <sup>≜</sup> xB y<sup>0</sup>

$$\mathbf{y} = \mathbf{S}b\left(\mathbf{q}(17, 19) \begin{array}{c} 17\\ \mathbf{Z}(\mathbf{x})\end{array}\right) = \mathbf{y}(19) \left[= \Phi[\mathbf{Q}(\mathbf{x}, \mathbf{Y})] \right] \quad \text{and} \quad \mathbf{y}' = \mathbf{S}b\left(\mathbf{y} \begin{array}{c} 19\\ \mathbf{Z}(\mathbf{y})\end{array}\right) \tag{15}$$

Any of the following notations can be used

$$\begin{aligned} q(u\_1, \mu\_2) &= q(17, 19) = \Phi[\mathbf{Q}(X, Y)] = \Phi[q(\mathbf{u}, \boldsymbol{\sigma})] = \Phi[\mathbf{Q}(X, Y)] \\ q(u\_1, \mu\_2) &= q(17, 19) = \Phi[\mathbf{Q}(X, Y)] \\ q(u\_1, \mu\_2), q(17, 19), q(\mu, \boldsymbol{\sigma}) &\triangleq \mathbf{Q}(X, Y), \quad \dots \\ q[\mathbf{Z}(\mathbf{x}), \mu\_2] &= y(\mu\_2) = q[\mathbf{Z}(\mathbf{x}), 19] = y(19) = y = \Phi[\mathbf{Q}(\mathbf{x}, Y)] \triangleq \mathbf{Q}(\mathbf{x}, Y) \end{aligned} \tag{16}$$

The following Admissible Substitution Sb 19 y ZðyÞ 0 @ 1 A is carried out in the second step of the

given Double Substitution Sb 17 19 q ZðxÞ ZðyÞ 0 @ 1 A; in the Goedel variable number qð17; 19Þ, we

first put 17 :¼ ZðxÞ and in the result q½ZðxÞ; 19� we put 19 :¼ ZðyÞ. Then

$$y' = y[Z(y)] = [y(19)]\_{19 = Z(y)} = q[Z(\mathbf{x}), Z(y)] = \Phi[Q(\mathbf{x}, y)] \triangleq Q(\mathbf{x}, y) \tag{17}$$

The CLAIM y<sup>0</sup> 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 <sup>11</sup> it creates from the given FORMULA y an infinite sequence of semantically identical FORMULAE y<sup>0</sup> ½¼ y½Zðy��, y½Zðy<sup>0</sup> Þ� ½¼ y½Z½y½Zðy����, … with the aim to end the process by 'printing' just the value y<sup>0</sup> . But it never reveals this outcome y<sup>0</sup> ; however, we —metatheoretically—know it. It never gets as far as to print the natural number y<sup>0</sup> 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<sup>0</sup> 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

<sup>11</sup>Caused by the application of the (Cantor) diagonal argument.

analyzer is trying to calculate and 'print' y<sup>0</sup> , 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 LT AP (and LP). This is the reason for not-recursivity of the relations Bewð�Þ; the upper limit of its computing process is missing. First, we have q½ZðxÞ,19� <sup>19</sup>:¼Zðy<sup>Þ</sup> ffi <sup>q</sup>½ZðxÞ;ZðyÞ� ¼ <sup>y</sup>½ZðyÞ� ¼ <sup>y</sup><sup>0</sup> and then <sup>0</sup> try<sup>0</sup> <sup>12</sup>

3. Goedel substitution function and FORMULA 17Gen r

FORMULA qð17; 19Þ ½the ARITHMETIZATION of QðX;YÞ�,

17

1

qðu1;u2Þ ¼ qð17; 19Þ ¼ Φ½QðX;YÞ�

q

0 @

y and then, it is the FORMULA y speaking about itself.

semantically identical FORMULAE y<sup>0</sup> ½¼ y½Zðy��, y½Zðy<sup>0</sup>

end the process by 'printing' just the value y<sup>0</sup>

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

qðu1;u2Þ; qð17; 19Þ; qðu;vÞ≜ QðX;YÞ; …

ZðxÞ

qð17; 19Þ

Any of the following notations can be used

The following Admissible Substitution Sb

given Double Substitution Sb

of the substitution 19 :¼ ZðyÞ

y ¼ Sb

0 @

286 Ontology in Information Science

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<sup>0</sup> that the Goedel (variable) number (his 'CLASS' SIGN) y arises from Admissible Substitution from the

qðu1;u2Þ ¼ qð17; 19Þ ¼ Φ½QðX;YÞ� ¼ Φ½qðu;vÞ� ¼ Φ½QðX;YÞ�

y

0 @

17 19

ZðxÞ ZðyÞ

first put 17 :¼ ZðxÞ and in the result q½ZðxÞ; 19� we put 19 :¼ ZðyÞ. Then

q½ZðxÞ;u2� ¼ yðu2Þ ¼ q½ZðxÞ; 19� ¼ yð19Þ ¼ y ¼ Φ½Qðx;YÞ�≜ Qðx;YÞ

19

1

<sup>y</sup><sup>0</sup> <sup>¼</sup> <sup>y</sup>½ZðyÞ� ¼ ½yð19Þ�<sup>19</sup>:¼Zðy<sup>Þ</sup> <sup>¼</sup> <sup>q</sup>½ZðxÞ;ZðyÞ� ¼ <sup>Φ</sup>½Qðx;yÞ�<sup>≜</sup> <sup>Q</sup>ðx;yÞ ð17<sup>Þ</sup>

<sup>11</sup> it creates from the given FORMULA y an infinite sequence of

. But it never reveals this outcome y<sup>0</sup>

ZðyÞ

1

The CLAIM y<sup>0</sup> 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

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

—metatheoretically—know it. It never gets as far as to print the natural number y<sup>0</sup> 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<sup>0</sup> 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

A ¼ yð19Þ ½¼ Φ½Qðx;YÞ� � and y<sup>0</sup> ¼ Sb

19

1

A ð15Þ

ð16Þ

ZðyÞ

y

0 @

A is carried out in the second step of the

Þ� ½¼ y½Z½y½Zðy����, … with the aim to

; however, we

A; in the Goedel variable number qð17; 19Þ, we

$$\begin{split} \mathcal{Y} &\cong q[\mathcal{Z}(\mathbf{z}), \mathcal{Z}[q[\mathcal{Z}(\mathbf{x}), 19]]]\_{19 = \mathcal{Z}(y)} \\ &\cong q[\mathcal{Z}(\mathbf{x}), \mathcal{Z}[q[\mathcal{Z}(\mathbf{x}), \mathcal{Z}(y)]]] = \mathcal{Y}[\mathcal{Z}[q[\mathcal{Z}(y)]]] \\ &\cong q[\mathcal{Z}(\mathbf{x}), \mathcal{Z}[q[\mathcal{Z}(\mathbf{x}), \mathcal{Z}[q[\mathcal{Z}(\mathbf{x}), 19]]]]]\_{19 = \mathcal{Z}(y)} \\ &\cong q[\mathcal{Z}(\mathbf{x}), \mathcal{Z}[q[\mathcal{Z}(\mathbf{x}), \mathcal{Z}[q[\mathcal{Z}(\mathbf{x}), \mathcal{Z}[q[\mathcal{Z}(\mathbf{x}), 19]]]]]]\_{19 = \mathcal{Z}(y)} \\ &\cong q[\mathcal{Z}(\mathbf{x}), \mathcal{Z}[q[\mathcal{Z}(\mathbf{x}), \mathcal{Z}[q[\mathcal{Z}(\mathbf{x}), \mathcal{Z}[q[\mathcal{Z}(\mathbf{x}), \mathcal{Z}(y)]]]]]] \\ &\cong q[\mathcal{Z}(\mathbf{x}), \mathcal{Z}[q[\mathcal{Z}(\mathbf{x}), \mathcal{Z}[q[\mathcal{Z}(\mathbf{x}), \mathcal{Z}[q[\mathcal{Z}(\mathbf{x}), 19]]]]]]]]\_{19 = \mathcal{Z}(y)} \cong \dots \text{ ad lib.} \end{split} \tag{18}$$

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

$$\begin{array}{l} Sb\begin{pmatrix}19\\y(19)\end{pmatrix}\stackrel{\text{Def}}{=} 0\ \&Sb\left(\begin{array}{c}19\\\text{Neg}[y(19)]\end{array}\right)\begin{aligned} 19\\\text{Neg}[y(19)]\\\text{Z}(y)\end{aligned}\end{cases}\qquad\begin{aligned} 19\\\text{S}\overline{b}\left(\begin{array}{c}19\\\text{Z}(y)\end{array}\right)\begin{aligned} 19\\\text{S}\overline{b}\left(\begin{array}{c}y(19)\\\text{Z}(y)\end{array}\right)\stackrel{\text{Def}}{=} 0\\\hline\hline\\\begin{aligned} q(\text{x},y) &\equiv \text{xB}\left[\begin{array}{c}\text{S}\overline{b}\left(\begin{array}{c}19\\\text{Z}(y)\end{array}\right)\end{array}\right]&=q[\text{Z}(\text{x}),\text{Z}(\text{y})]=\text{y}\end{aligned}\end{cases}\end{cases}\qquad\begin{aligned} \overline{\begin{aligned} 19\\\text{Y}\end{bmatrix}\stackrel{\text{Def}}{=} 0\\\hline\hline\hline\\\begin{aligned} q(\text{x},y) &\equiv \text{y}\end{aligned}\end{cases}\qquad\begin{aligned} \overline{\begin{aligned} 19\\\text{Y}\end{bmatrix}\stackrel{\text{Def}}{=} 0\\\hline\hline\hline\\\begin{aligned} q(\text{x},y) &\equiv \text{z}\end{aligned}\end{aligned}\qquad\begin{aligned} \overline{\begin{aligned} 19\\\text{Z}(y)\end{bmatrix}\stackrel{\text{Def}}{=} 0\\\hline\hline\hline\\\begin{aligned} q(\text{x},y) &\equiv \text{z}\end{aligned}\end{aligned}\qquad\begin{aligned} \overline{\begin{aligned} q(\text{y}) &\in\mathbb{Z}\left(\begin{aligned} 19\\\text{Z}(y)\end{aligned}\end{aligned}\end{cases}$$

Also see the Proposition V in Refs. [3–5]. The mere grammar derivation, writability convenient to the recursive grammar is quite different from the T PA-provability. The Goedel number y<sup>0</sup> , 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<sup>0</sup> Þ ¼ 0�. It is true for the given x and it 'says': 'I, FORMULA y½ZðyÞ�, am in the system P

<sup>12</sup>By substitution 19 :<sup>¼</sup> <sup>Z</sup>ðy<sup>Þ</sup> nothing changes in variability of FORMULA y<sup>0</sup> by the VARIABLE 19. The number <sup>y</sup><sup>0</sup> 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.

<sup>r</sup>½ZðxÞ� ¼ <sup>r</sup>ð17Þ17:¼Zð<sup>x</sup>Þ¼ <sup>q</sup>½17;ZðpÞ�17:¼Zðx<sup>Þ</sup>

¼ q½ZðxÞ; Z½17Gen qð17; 19Þ�� ¼ q½ZðxÞ; ZðpÞ�� ¼ q½ZðpÞ� ¼ q<sup>0</sup>

ð22Þ

289

ð23Þ

¼ q½ZðxÞ, Z½Φ½u1Πqðu1, u2Þ��� ¼ Φ½Q½x, Φ½∀<sup>x</sup><sup>∈</sup> <sup>X</sup>jQðx, Y���

¼ ZðxÞGen q½ZðxÞ;Z½17Gen qð17; 19Þ�� ¼ p½ZðpÞ� ¼ p<sup>0</sup>

¼ 17Gen q½17;Z½pð19Þ�� ¼ 17Gen q½17;Z½17Gen qð17; 19��

≜ Q½X;Φ½QðX;YÞ�� ¼ Q½ℕ0;Φ½Qðℕ0;Y��

Information Transfer and Thermodynamic Point of View on Goedel Proof

http://dx.doi.org/10.5772/intechopen.68809

ffi 17Gen q½17;Z½17Gen qð17; 19Þ�� ¼ 17Gen rð17Þ

¼ Φ½∀<sup>x</sup><sup>∈</sup> <sup>X</sup>jΦ½Q½x;Φ½∀<sup>x</sup><sup>∈</sup> <sup>X</sup>jQðx;YÞ���� ¼ 17Gen r

The relation QðX;pÞ, QðX;pÞ ¼ ∀<sup>x</sup><sup>∈</sup> <sup>X</sup>jQ½x;Φ½∀<sup>x</sup><sup>∈</sup> <sup>X</sup>Qðx;p�� 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

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<sup>P</sup>� than the language LP of the system P and so outside both of the language LP (and as such, outside of the language LT 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.<sup>15</sup> 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<sup>P</sup>� 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

Let the Goedel number t½ZðxÞ;ZðyÞ� be DESCRIPTION of the mechanism of the transfer y from

15We have, inside of them, only N <sup>0</sup> 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 <sup>1</sup> containing, as its

elements, the N 0-sets; thus it can speak about the theory TPA, N <sup>0</sup> < N <sup>1</sup> and cannot be in it or in the system P.

¼ Φ½Q½x, Φ½QðX, YÞ��� ¼ Φ½Q½x, Φ½Qðℕ0, Y���

With regard of quantification r½ZðxÞ� over values ZðxÞ of the variable u1, we write

¼ Φ½u1Π½Φ½q½u1;Φ½u1Πqðu1;u2�����

<sup>≜</sup> <sup>Q</sup>ðX;pÞ ¼ <sup>Q</sup>ðX;YÞ<sup>Y</sup>:¼<sup>p</sup>

point of view, it is an arithmetic code but of the not-arithmetic claim.<sup>16</sup>

x (on the level of the system P and the theory T PA) in the channel K,

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

of the system P itself end up in the infinite cycle.

ZðxÞGen r½ZðxÞ� ¼ ZðxÞGen q½ZðxÞ;ZðpÞ�� ¼ ZðxÞGen p½ZðpÞ� ¼ ZðxÞGen q<sup>0</sup>

#### 3.1. FORMULA 17Gen r and information transfer

With regard of the fact that FORMULA y<sup>0</sup> 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 ℕ0. Then, within the framework of the theory T PA, we put 17Gen y<sup>0</sup> ¼ Def 0 and thus <sup>J</sup>ð17Gen y<sup>0</sup> Þ ¼Def 0.13 In the proof we put p : 17Gen q, ½17 ffi u<sup>1</sup> ≜ X, 19 ≜u<sup>2</sup> ≜Y, q ¼ qð17, 19Þ�, and then, in compliance with the Goedel notation,

$$p = 17 \text{Gen} \neq (17, 19) = \Phi[\boldsymbol{\mu}\_1 \boldsymbol{\Pi} \boldsymbol{q}(\boldsymbol{\mu}\_1, \boldsymbol{\mu}\_2)] \qquad [= \Phi[\boldsymbol{\forall}\_{\boldsymbol{x} \in \mathbf{X}} | \boldsymbol{Q}(\boldsymbol{x}, \boldsymbol{Y})] \triangleq \boldsymbol{Q}(\mathbf{X}, \boldsymbol{Y})] \triangleq \boldsymbol{Q}(\mathbb{N}\_0, \boldsymbol{Y})\qquad(20)$$

The metalanguage symbol QðX, YÞ in (20) or the symbol Qðℕ0,YÞ is read as follows:

'None x∈ Xðℕ0Þ is in the relation INFERENCE to the content (to the selective space Y) of the variable Y. From any given x, x ¼ Φð a !Þ ¼ <sup>Φ</sup>ð½ak <sup>1</sup> , akþ<sup>1</sup>�Þ ��! , x ∈ Xðℕ0Þ, any Goedel number Φðakþ<sup>1</sup>Þ 6¼ 0, writable as the proposed outcome of the INFERENCE from the given x, is NOT INFERRED in reality.'

$$\text{We put } r := Sb \begin{pmatrix} 19 \\ q \\ Z(p) \end{pmatrix} = Sb \left( q(17, 19) \begin{array}{c} 19 \\\\ Z(p) \end{array} \right) = Sb \left( q(17, 19) \begin{array}{c} 19 \\\\ Z[17 \text{Gen } q(17, 19)] \end{array} \right).$$

The Goedel number r, r ≜rð17Þ ¼ Φ½QðX, pÞ� is, by the substitution ZðpÞ, supposingly [3–5], 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, ½q½17, Z½17Gen qð17, 19��,

r ¼ rð17Þ ¼ q½17, Z½pð19Þ�� ¼ q½17, Z½17Gen qð17, 19��≜ q½u1, ZðpÞ�≜ QðX, pÞ ¼ q½u1, Φ½u1Πqðu1, u2Þ�� ¼ Φ½Q½X, Φ½∀<sup>x</sup><sup>∈</sup> <sup>X</sup>jQðx, Y���, <sup>≜</sup> <sup>Q</sup>ðX, pÞ ¼ <sup>Q</sup>ðX,YÞ<sup>Y</sup>:¼<sup>p</sup> ≜ Q½X, Φ½QðX, Y��≜ Q½X, Φ½Qðℕ0, Y�� ð21Þ

Further<sup>14</sup> <sup>Q</sup>ðX,YÞ<sup>X</sup>:¼<sup>x</sup> <sup>¼</sup> <sup>Q</sup>ðx, YÞ, Qðx,YÞ<sup>Y</sup>:¼<sup>p</sup> <sup>¼</sup> <sup>Q</sup>ðx, p<sup>Þ</sup> and then,

nor on the sequence of operations Sb � � � 0 @ 1 Aand ½�� Gen ½���.

<sup>13</sup>From that y<sup>0</sup> is NOT INFERRED follows its NOT-INFERRABILITY/NOT-PROVABILITY.

<sup>14</sup>And similarly for <sup>Q</sup>ðX;YÞ<sup>Y</sup>:¼<sup>p</sup> <sup>¼</sup> <sup>Q</sup>ðX;pÞ, <sup>Q</sup>ðX;pÞ<sup>X</sup>:¼<sup>x</sup> <sup>¼</sup> <sup>Q</sup>ðx;pÞ. It depends neither on the sequence of substitution steps

$$\begin{aligned} r[Z(\mathbf{x})] &= r(\mathbf{1}\mathbf{7})\_{\mathbf{1}\mathbf{7} \coloneqq Z(\mathbf{x})} = q[\mathbf{1}\mathbf{7}, \mathbf{Z}(p)]\_{\mathbf{1}\mathbf{7} \coloneqq Z(\mathbf{x})} \\ &= q[Z(\mathbf{x}), \mathbf{Z}[\mathbf{1}\mathbf{7}\mathbf{G}\text{en } q(\mathbf{1}\mathbf{7}, \mathbf{1}\mathbf{9})]] = q[Z(\mathbf{x}), Z(p)] \ = q[Z(p)] = q' \\ &= q[Z(\mathbf{x}), \mathbf{Z}[\Phi[u\_1\mathbf{1}\mathbf{I}q(u\_1, u\_2)]]] = \Phi[Q[\mathbf{x}, \Phi[\mathbf{V}\_{\mathbf{x}\in\mathbf{X}}]Q(\mathbf{x}, \mathbf{Y})]] \\ &= \Phi[Q[\mathbf{x}, \Phi[Q(\mathbf{X}, \mathbf{Y})]]] = \Phi[Q[\mathbf{x}, \Phi[Q(\mathbf{N\_0}, \mathbf{Y})]]] \end{aligned} \tag{22}$$

With regard of quantification r½ZðxÞ� over values ZðxÞ of the variable u1, we write

(by it means) from the Goedel number x UNPROVABLE.' And, by this, it also states both the

With regard of the fact that FORMULA y<sup>0</sup> 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 ℕ0. Then,

the proof we put p : 17Gen q, ½17 ffi u<sup>1</sup> ≜ X, 19 ≜u<sup>2</sup> ≜Y, q ¼ qð17, 19Þ�, and then, in compliance

'None x∈ Xðℕ0Þ is in the relation INFERENCE to the content (to the selective space Y) of the

!Þ ¼ <sup>Φ</sup>ð½ak

Φðakþ<sup>1</sup>Þ 6¼ 0, writable as the proposed outcome of the INFERENCE from the given x, is NOT

19

1 A ¼ Sb

ZðpÞ

The Goedel number r, r ≜rð17Þ ¼ Φ½QðX, pÞ� is, by the substitution ZðpÞ, supposingly [3–5], 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

r ¼ rð17Þ ¼ q½17, Z½pð19Þ�� ¼ q½17, Z½17Gen qð17, 19��≜ q½u1, ZðpÞ�≜ QðX, pÞ

14And similarly for <sup>Q</sup>ðX;YÞ<sup>Y</sup>:¼<sup>p</sup> <sup>¼</sup> <sup>Q</sup>ðX;pÞ, <sup>Q</sup>ðX;pÞ<sup>X</sup>:¼<sup>x</sup> <sup>¼</sup> <sup>Q</sup>ðx;pÞ. It depends neither on the sequence of substitution steps

¼ q½u1, Φ½u1Πqðu1, u2Þ�� ¼ Φ½Q½X, Φ½∀<sup>x</sup><sup>∈</sup> <sup>X</sup>jQðx, Y���,

<sup>1</sup> , akþ<sup>1</sup>�Þ ��!

0 @

The metalanguage symbol QðX, YÞ in (20) or the symbol Qðℕ0,YÞ is read as follows:

qð17, 19Þ

0 @

<sup>≜</sup> <sup>Q</sup>ðX, pÞ ¼ <sup>Q</sup>ðX,YÞ<sup>Y</sup>:¼<sup>p</sup>

Further<sup>14</sup> <sup>Q</sup>ðX,YÞ<sup>X</sup>:¼<sup>x</sup> <sup>¼</sup> <sup>Q</sup>ðx, YÞ, Qðx,YÞ<sup>Y</sup>:¼<sup>p</sup> <sup>¼</sup> <sup>Q</sup>ðx, p<sup>Þ</sup> and then,

13From that y<sup>0</sup> is NOT INFERRED follows its NOT-INFERRABILITY/NOT-PROVABILITY.

1

Aand ½�� Gen ½���.

� � �

0 @

p ¼ 17Gen qð17, 19Þ ¼ Φ½u1Πqðu1, u2Þ� ½¼ Φ½∀<sup>x</sup><sup>∈</sup> <sup>X</sup>jQðx, YÞ�≜ QðX, YÞ�≜ Qðℕ0, YÞ ð20Þ

Def 0 and thus <sup>J</sup>ð17Gen y<sup>0</sup>

, x ∈ Xðℕ0Þ, any Goedel number

Z½17Gen qð17, 19Þ�

1 A.

ð21Þ

19

qð17, 19Þ

≜ Q½X, Φ½QðX, Y��≜ Q½X, Φ½Qðℕ0, Y��

Þ ¼ Def 0.13 In

property of the system P and the theory T PA.

variable Y. From any given x, x ¼ Φð a

19

1 A ¼ Sb

ZðpÞ

BOUND in it, ½q½17, Z½17Gen qð17, 19��,

nor on the sequence of operations Sb

q

0 @

with the Goedel notation,

288 Ontology in Information Science

INFERRED in reality.'

We put r :¼ Sb

3.1. FORMULA 17Gen r and information transfer

within the framework of the theory T PA, we put 17Gen y<sup>0</sup> ¼

$$\begin{aligned} Z(\mathbf{x})\text{Gen }r[Z(\mathbf{x})] &= Z(\mathbf{x})\text{Gen }q[Z(\mathbf{x}), Z(p)]] = Z(\mathbf{x})\text{Gen }p[Z(p)] = Z(\mathbf{x})\text{Gen }q' \\ &= Z(\mathbf{x})\text{Gen }q[Z(\mathbf{x}), Z[17\text{Gen }q(17, 19)]] = p[Z(p)] = p' \end{aligned}$$

$$\begin{aligned} &\cong 17\text{Gen }q[17, 2[17\text{Gen }q(17, 19)]] = 17\text{Gen }r(17) \\ &= 17\text{Gen }q[17, 2[p(19)]] = 17\text{Gen }q[17, 2[17\text{Gen }q(17, 19)]] \\ &= \Phi[u\_1\Pi[\Phi[q[u\_1, \Phi[u\_1\Pi q(u\_1, u\_2)]]]] \\ &= \Phi[\Psi\_{\mathbf{x}\in\mathbf{X}}\mathbf{x}]\Phi[\mathbf{Q}[\mathbf{x}, \Phi[\mathbf{x}\_{\mathbf{x}}\mathbf{x}] \mathbf{Q}(\mathbf{x}, \mathbf{y})]]] = 17\text{Gen }r \\ &\triangleq \Phi(\mathbf{X}\cdot\mathbf{y}) = \mathbf{Q}(\mathbf{X}, \mathbf{Y})\_{\mathbf{Y}=\mathbf{y}} \triangleq \mathbf{Q}[\mathbf{X}, \Phi[\mathbf{Q}(\mathbf{X}, \mathbf{Y})]] = \mathbf{Q}[\mathbf{N}\_{\Phi}\Phi[\mathbf{Q}(\mathbf{N}\_{\mathbf{U}}, \mathbf{Y})]] \end{aligned}$$

The relation QðX;pÞ, QðX;pÞ ¼ ∀<sup>x</sup><sup>∈</sup> <sup>X</sup>jQ½x;Φ½∀<sup>x</sup><sup>∈</sup> <sup>X</sup>Qðx;p�� 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<sup>P</sup>� than the language LP of the system P and so outside both of the language LP (and as such, outside of the language LT 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.<sup>15</sup> 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<sup>P</sup>� 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.<sup>16</sup>

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,

<sup>15</sup>We have, inside of them, only N <sup>0</sup> 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 <sup>1</sup> containing, as its elements, the N 0-sets; thus it can speak about the theory TPA, N <sup>0</sup> < N <sup>1</sup> and cannot be in it or in the system P. 16Thus it is not a common number as the [3–5] claims and neither is r.

$$\operatorname{Sb}\begin{pmatrix} 17 & 19\\ t & Z(\mathbf{x}) \end{pmatrix} \cong \operatorname{Subst} \operatorname{t}^{\mathbb{C}}(\operatorname{U}\_{1}, \operatorname{U}\_{2}) \begin{bmatrix} \operatorname{U}\_{1} & \operatorname{U}\_{2} \\ \operatorname{J}(\mathbf{x}) & \operatorname{J}(\mathbf{y}) \end{bmatrix} \equiv [\operatorname{J}(\mathbf{x}) - \operatorname{J}(\mathbf{y}) \neq \operatorname{J}(\mathbf{x})] \tag{24}$$

∃<sup>x</sup><sup>∈</sup> <sup>X</sup>jt

equivalent informationally. It is valid that Jðp<sup>0</sup>

h i � <sup>∀</sup><sup>x</sup><sup>∈</sup> <sup>X</sup>j½JðxÞ � <sup>J</sup>ðxjp<sup>0</sup>

<sup>Þ</sup>∗17Gen r <sup>¼</sup> <sup>Φ</sup>ðak

<sup>Þ</sup>∗Φð0Þ� ¼ <sup>J</sup>½Φð½a<sup>k</sup>

Þ� ffi ½JðXjp<sup>0</sup>

1 ! Þ∗Sb

∀<sup>x</sup><sup>∈</sup> <sup>X</sup>j½JðxÞ ¼ Jðxjp<sup>0</sup>

Þ

1 !

Þ ¼ JðnullÞ ¼ 0, JðyÞ ¼ Jðp<sup>0</sup>

has not been INFERRED either.

18Attention (!) but <sup>x</sup> contains the message <sup>p</sup> that <sup>J</sup>ðxÞ ¼ <sup>J</sup>ðxjpÞ.

∀<sup>x</sup><sup>∈</sup> <sup>X</sup>jτðx;p<sup>0</sup>

<sup>x</sup> <sup>¼</sup> <sup>Φ</sup>ðak 1 !

<sup>J</sup>ðxÞ ¼ <sup>J</sup>½Φðak

valid [14, 16] that

Jðxjp<sup>0</sup>

<sup>K</sup>½JðxÞ;J½∃<sup>x</sup><sup>∈</sup> <sup>X</sup><sup>t</sup>

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<sup>0</sup> and on interrupted.' Or, we have made the interrupted transfer channel directly by this p<sup>0</sup> when we assumed it belonged to the set of

we want to transfer this p<sup>0</sup> from the input x which includes this p<sup>0</sup> (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)

> Þ > 0� h i � <sup>∃</sup><sup>x</sup>∈ℕ<sup>0</sup> j½JðxÞ � <sup>J</sup>ðxjp<sup>0</sup>

> > 1

CCA <sup>¼</sup> <sup>Φ</sup>ð½ak

� ¼ <sup>J</sup>ðxjpÞ ¼ <sup>J</sup>½Φð½ak

The channel K, however, always works only with the not zero and the positive difference of information amounts JðxÞ � JðxjyÞ and in the theory T PA now it is valid that JðyÞ ¼ JðxÞ�

erroneous. No input message x having a relation to the output message p<sup>0</sup> exists. The FOR-MULA 17Gen r both creates and describes behavior of the not functioning (interrupted) information transfer, from p<sup>0</sup> on further. For the efficiency η of the information transfer, it is then

> <sup>η</sup> <sup>¼</sup> <sup>J</sup>ðp<sup>Þ</sup> JðXÞ

• No information transfer channel K transfers its (internal) state xjy ½the information JðxjyÞ�

• There is no x ∈ℕ<sup>0</sup> for which it is possible to generate the Goedel number Φ½Qðℕ0;YÞ� which claims that there is no x ∈ℕ<sup>0</sup> 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 Φðakþ<sup>1</sup>Þ ¼ 17Gen r

19

ZðpÞ

p

0

BB@

<sup>1</sup>�Þ � 20

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

given as its input message x, it behaves as interrupted.

messages transferrable from the source X. So, first we interrupt the channel K for p<sup>0</sup>

<sup>K</sup>½JðxÞ;Jðy��� <sup>&</sup>gt; <sup>0</sup> � <sup>T</sup>ðX;y<sup>Þ</sup> <sup>&</sup>gt; <sup>0</sup> � <sup>Q</sup>ðX;yÞ ð27<sup>Þ</sup>

Information Transfer and Thermodynamic Point of View on Goedel Proof

Þ ¼ 0 for any x, x∈ X ½so ∀<sup>c</sup> <sup>∈</sup> <sup>X</sup>j½Jðp<sup>0</sup>

Þ ¼ JðXÞ > 0� and for the simplicity is Jðp<sup>0</sup>

1�, p<sup>0</sup>

, and then,

291

ð28Þ

Þ ¼ 0��,

jXÞ ¼ 0

Þ > 0�

http://dx.doi.org/10.5772/intechopen.68809

<sup>1</sup>�, nullÞ

¼ 0 ð29Þ

<sup>1</sup>�Þ

h i <sup>¼</sup> "1"

Þ ffi <sup>Φ</sup>ð½ak

<sup>1</sup>�Þ�, xj<sup>p</sup> <sup>¼</sup> <sup>Φ</sup>ð½ak

Þ ¼ <sup>J</sup>ðnullÞ ¼ 018. It means that our assumption about <sup>p</sup><sup>0</sup> ½¼ <sup>r</sup>� is

But, when it is valid that Sb 19 y ZðyÞ 0 @ 1 A ¼ 0 ¼ Sb 17 19 q ZðxÞ ZðyÞ 0 @ 1 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. [3–5], 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<sup>0</sup> ¼ p½ZðpÞ� is valid

$$\begin{split}p' & \stackrel{\text{Def}}{=} \text{Subst}\ p(\mathcal{U}\_2) \binom{\mathcal{U}\_2}{l(p)} = \text{Subst}\ \mathcal{U}\_1 \text{Gen}\ q^\p(\mathcal{U}\_1, \mathcal{U}\_2) \binom{\mathcal{U}\_2}{l[\mathcal{U}\_1 \text{Gen}\ q^\p(\mathcal{U}\_1, \mathcal{U}\_2)]} \\ p' &= 17 \text{Gen}\ q^\p[\mathcal{U}\_1, l[\mathcal{U}\_1 \text{Gen}\ q^\p(\mathcal{U}\_1, \mathcal{U}\_2)]] = \mathcal{U}\_1 \text{Gen}\ q^\p[\mathcal{U}\_1, l(p)] = p[\mathcal{I}(p)] \\ & = \mathcal{U}\_1 \text{Gen}\ r(\mathcal{U}\_1) \ = \mathcal{U}\_1 \text{Gen}\ r \\ & \stackrel{\text{def}}{=} u\_1 \Pi[q^\p[\mathcal{U}\_1, l[u\_1 \Pi q^\k(u\_1, u\_2)]]] = \Psi\_{\mathbf{x} \in X}[\mathbf{Q}^\k[\mathcal{X}, \mathbf{Q}[\Psi\_{\mathbf{x} \in X} | \mathbf{Q}^{\k}(\mathbf{x}, \mathcal{Y})]] \\ & = \mathbf{Q}^{\k}(\mathcal{X}, p)] = \mathbf{Q}^{\k}(\mathcal{X}, \mathcal{Y})\_{\mathcal{Y} := p} = \mathbf{Q}^{\k}[\mathcal{X}, \Psi[\mathbf{Q}^{\k}(\mathcal{X}, \mathcal{Y})]] = \mathbf{Q}^{\k}[\mathbb{N}\_{\emptyset}, \Psi[\mathbf{Q}^{\k}(\mathbb{N}\_{\emptyset}, \mathcal{Y})]] \end{split} \tag{25}$$

So, the message p<sup>0</sup> (the message p about itself) is not-transferrable from any message x,

$$\overline{\mathbb{P}[\mathbf{x}\overline{\mathbb{B}^{[\mathbb{K}]}}\ p'=\text{''I''}]} \equiv [\overline{\mathbf{x}\overline{\mathbb{B}^{[\mathbb{K}]}}\ p}=\text{''I''}] \equiv [\pi^{[\mathbb{K}]}(\mathbf{x},\mathbf{y})=\text{''0''}] \equiv [\mathbb{J}(p)=\mathbf{0}] \equiv [\mathbb{J}(p')=\mathbf{0}] \tag{26}$$

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 <sup>K</sup> as well.17 Its 'errorness' is in our awaiting of the non-zero outcome <sup>J</sup>ðy<sup>Þ</sup> <sup>&</sup>gt; 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 akþ<sup>1</sup> ½CLAIM y, y ¼ Φðakþ<sup>1</sup>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),

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

$$\overline{\exists\_{\mathbf{x}\in\mathcal{X}}|t^{\mathcal{K}}[f(\mathbf{x}),\mathbf{J}[\exists\_{\mathbf{x}\in\mathcal{X}}t^{\mathcal{K}}[f(\mathbf{x}),\mathbf{J}(\mathbf{y})]]| > 0} \equiv \overline{T(\mathbb{X},\mathcal{y}) > 0} \equiv \mathcal{Q}(\mathbb{X},\mathbf{y}) \tag{27}$$

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<sup>0</sup> and on interrupted.' Or, we have made the interrupted transfer channel directly by this p<sup>0</sup> when we assumed it belonged to the set of messages transferrable from the source X. So, first we interrupt the channel K for p<sup>0</sup> , and then, we want to transfer this p<sup>0</sup> from the input x which includes this p<sup>0</sup> (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<sup>0</sup> Þ ¼ 0 for any x, x∈ X ½so ∀<sup>c</sup> <sup>∈</sup> <sup>X</sup>j½Jðp<sup>0</sup> Þ ¼ 0��,

Sb

290 Ontology in Information Science

t

But, when it is valid that Sb

p<sup>0</sup> ¼

ble, it is),

DefSubst pðU2<sup>Þ</sup>

¼ U1Gen rðU1Þ ¼ U1Gen r

<sup>½</sup>xB½K� <sup>p</sup><sup>0</sup> <sup>¼</sup> "1"��½xB½K� <sup>p</sup> <sup>¼</sup> "1"��½τ½K�

17Gen r or 'SENTENTIAL FORMULA'/'PROPOSITION.'

0 @

17 19

1

y

0 @

U<sup>2</sup> JðpÞ

!

19

1

ZðyÞ

<sup>A</sup> ffi Subst t<sup>K</sup>ðU1, U2<sup>Þ</sup> <sup>U</sup><sup>1</sup> <sup>U</sup><sup>2</sup>

A ¼ 0 ¼ Sb

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. [3–5], 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<sup>0</sup> ¼ p½ZðpÞ� is valid

<sup>¼</sup> Subst U1Gen q<sup>K</sup>ðU1, U2<sup>Þ</sup>

<sup>¼</sup> <sup>Q</sup><sup>K</sup>ðX, pÞ� ¼ <sup>Q</sup><sup>K</sup>ðX,YÞ<sup>Y</sup>:¼<sup>p</sup> <sup>¼</sup> <sup>Q</sup><sup>K</sup>½X, <sup>Φ</sup>½Q<sup>K</sup>ðX, YÞ�� ¼ <sup>Q</sup><sup>K</sup>½ℕ0, <sup>Φ</sup>½Q<sup>K</sup>ðℕ0, Y��

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 <sup>K</sup> as well.17 Its 'errorness' is in our awaiting of the non-zero outcome <sup>J</sup>ðy<sup>Þ</sup> <sup>&</sup>gt; 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

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 akþ<sup>1</sup> ½CLAIM y, y ¼ Φðakþ<sup>1</sup>Þ ¼ 17Gen r� is not inferable (INFERABLE) in the given inferential system P (but in another one making its construction-INFERENCE possi-

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

<sup>p</sup><sup>0</sup> <sup>¼</sup> <sup>17</sup>Gen qK½U1, J½U1Gen qKðU1, U2Þ�� ¼ <sup>U</sup>1Gen qK½U1, JðpÞ� ¼ <sup>p</sup>½JðpÞ�

ffi <sup>u</sup>1Π½q<sup>K</sup>½u1, J½u1Πq<sup>K</sup>ðu1, u2Þ���� ¼ <sup>∀</sup><sup>x</sup><sup>∈</sup> <sup>X</sup>jQ<sup>K</sup>½X, <sup>Φ</sup>½∀<sup>x</sup><sup>∈</sup> <sup>X</sup>jQ<sup>K</sup>ðx, Y��

So, the message p<sup>0</sup> (the message p about itself) is not-transferrable from any message x,

JðxÞ JðyÞ � �

q

0 @

17 19

1

U<sup>2</sup> <sup>J</sup>½U1Gen q<sup>K</sup>ðU1, U2Þ�

!

ZðxÞ ZðyÞ

ðx;yÞ ¼ "0"��½JðpÞ ¼ 0��½Jðp<sup>0</sup>

� ½JðxÞ � JðyÞ 6¼ JðxÞ� ð24Þ

A then the number yis not

ð25Þ

Þ ¼ 0� ð26Þ

ZðxÞ ZðyÞ

$$\begin{aligned} \forall\_{\mathbf{x}\in\mathcal{X}}[I(\mathbf{x}) = f(\mathbf{x}|\mathbf{p}')] \cong \left[f(\mathbf{X}|\mathbf{p}') = f(\mathbf{X}) > 0\right] \text{ and for the simplicity is } I(p'|\mathbf{X}) = 0\\ \left[\forall\_{\mathbf{x}\in\mathcal{X}}[\pi(\mathbf{x}, p')] \equiv \left[\forall\_{\mathbf{x}\in\mathcal{X}}[\overline{I(\mathbf{x})} - f(\mathbf{x}|p') > 0]\right] \equiv \left[\exists\_{\mathbf{x}\in\mathbb{N}\_{0}}[I(\mathbf{x}) - f(\mathbf{x}|p') > 0]\right] \equiv \text{"I"} \end{aligned} \tag{28}$$

$$\mathbf{x} = \Phi(\overrightarrow{a\_{1}^{k}}) \* 17 \text{Gen } r = \Phi(\overrightarrow{a\_{1}^{k}}) \* \mathbf{S} \mathbf{b} \begin{pmatrix} 19\\p\\Z(p) \end{pmatrix} = \Phi([\overrightarrow{a\_{1}^{k}}], p') \cong \Phi([\overrightarrow{a\_{1}^{k}}], \text{null}) \tag{28}$$

$$\begin{aligned} \mathbf{J}(\mathbf{x}) = \mathbf{J}[\Phi(\overrightarrow{a\_{1}^{k}}) \* \Phi(\mathbf{0})] = \mathbf{J}[\Phi([\overrightarrow{a\_{1}^{k}}]) \cdot \mathbf{2}^{0}] = \mathbf{J}(\mathbf{x}|p) = \mathbf{J}[\Phi([\overrightarrow{a\_{1}^{k}}])], \text{ } \mathbf{x}|p = \Phi([\overrightarrow{a\_{1}^{k}}]) \end{aligned} \tag{29}$$

The channel K, however, always works only with the not zero and the positive difference of information amounts JðxÞ � JðxjyÞ and in the theory T PA now it is valid that JðyÞ ¼ JðxÞ� Jðxjp<sup>0</sup> Þ ¼ JðnullÞ ¼ 0, JðyÞ ¼ Jðp<sup>0</sup> Þ ¼ <sup>J</sup>ðnullÞ ¼ 018. It means that our assumption about <sup>p</sup><sup>0</sup> ½¼ <sup>r</sup>� is erroneous. No input message x having a relation to the output message p<sup>0</sup> exists. The FOR-MULA 17Gen r both creates and describes behavior of the not functioning (interrupted) information transfer, from p<sup>0</sup> on further. For the efficiency η of the information transfer, it is then valid [14, 16] that

$$\eta = \frac{J(p)}{J(\mathbb{X})} = 0 \tag{29}$$

<sup>1</sup>�Þ

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

1

<sup>18</sup>Attention (!) but <sup>x</sup> contains the message <sup>p</sup> that <sup>J</sup>ðxÞ ¼ <sup>J</sup>ðxjpÞ.

## The metaarithmetic sense of the CLAIM ('SENTENTIAL PROPOSITION') 17Gen r is:

6¼

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 <sup>P</sup><sup>19</sup> and neither is <sup>r</sup>. The same is for Negð17Gen r<sup>Þ</sup> (it cannot be inferred in <sup>P</sup> for is not inferable in P.) Nevertheless, we are in accordance with the intuitive and obviously intended sense of the Goedel Proposition VI<sup>20</sup> 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 (methodo-

it is valid [?]. This equation describes the mutual relations among information entropies [(aver-

The quantities HðXÞ, HðYÞ, HðXjYÞ and HðYjXÞ are the input, the output, the loss and the noise

The difference HðXÞ � HðXjYÞ or the difference HðYÞ � HðYjXÞ defines the transinformation

When the channel K transfers the information (entropy) HðXÞ, but now just at the value of the

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

20Because, 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 ….'

HðXÞ � HðXjYÞ≜ TðX; YÞ ¼ TðY; XÞ≜ HðYÞ � HðYjXÞ ð31Þ

TðX; YÞ ¼ 0 ½� ð ¼ HðYÞ � HðYjXÞ 32Þ

HðXÞ � HðXjYÞ ¼ HðYÞ � HðYjXÞ ð30Þ

Information Transfer and Thermodynamic Point of View on Goedel Proof

http://dx.doi.org/10.5772/intechopen.68809

293

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

In any information transfer channel K the channel equation

TðX; YÞ or the transinformation TðY; XÞ, respectively,

• For HðYjXÞ ¼ 0, we have TðX; YÞ ¼ HðYÞ ¼ 0. • For HðYjXÞ 6¼ 0 we have HðYÞ ¼ HðYjXÞ 6¼ 0

simplicity, that HðYjXÞ ¼ 0.

19In the contrast to Refs. [3–5].

entropy HðXjYÞ, HðXÞ ¼ HðXjYÞ, then, necessarily, must be valid

age) information amounts] in the channel K.

logical) statements.

5. Appendix

entropy.

## 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 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 <sup>P</sup><sup>19</sup> and neither is <sup>r</sup>. The same is for Negð17Gen r<sup>Þ</sup> (it cannot be inferred in <sup>P</sup> for is not inferable in P.) Nevertheless, we are in accordance with the intuitive and obviously intended sense of the Goedel Proposition VI<sup>20</sup> 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.

## 5. Appendix

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

inconsistent system <sup>P</sup><sup>∗</sup>, to its part <sup>P</sup><sup>∗</sup> � <sup>P</sup> (P∗⊃

or, even 'In the consistent (?) theory exists ….'

is not identical with it nor with the system in their totality.

4. Conclusion

292 Ontology in Information Science

• Within the general formulative language L<sup>P</sup>� of the inconsistent metasystem P<sup>∗</sup> (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

> 6¼ P).

• So, the Goedel Proposition VI… (1931) [3–5] 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 …

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

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

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

In any information transfer channel K the channel equation

$$H(X) - H(X|Y) = H(Y) - H(Y|X) \tag{30}$$

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.

The difference HðXÞ � HðXjYÞ or the difference HðYÞ � HðYjXÞ defines the transinformation TðX; YÞ or the transinformation TðY; XÞ, respectively,

$$H(\mathbf{X}) - H(\mathbf{X}|\mathbf{Y}) \triangleq T(\mathbf{X}; \mathbf{Y}) \quad = \quad T(\mathbf{Y}; \mathbf{X}) \triangleq H(\mathbf{Y}) - H(\mathbf{Y}|\mathbf{X}) \tag{31}$$