Open access peer-reviewed chapter

Common Gnoseological Meaning of Gödel and Caratheodory Theorems

By Bohdan Hejna

Submitted: February 5th 2019Reviewed: June 11th 2019Published: October 30th 2019

DOI: 10.5772/intechopen.87975

Downloaded: 108

Abstract

We will demonstrate that the I. and the II. Caratheodory theorems and their common formulation as the II. Law of Thermodynamics are physically analogous with the real sense of the Gödel’s wording of his I. and II. incompleteness theorems. By using physical terms of the adiabatic changes the Caratheodory theorems express the properties of the Peano Arithmetic inferential process (and even properties of any deductive and recursively axiomatic inference generally); as such, they set the physical and then logical limits of any real inference (of the sound, not paradoxical thinking), which can run only on a physical/thermodynamic basis having been compared with, or translated into the formulations of the Gödel’s proof, they represent the first historical and clear statement of gnoseological limitations of the deductive and recursively axiomatic inference and sound thinking generally. We show that semantically understood and with the language of logic and meta-arithmetics, the full meaning of the Gödel proof expresses the universal validity of the II. law of thermodynamics and that the Peano arithmetics is not self-referential and is consistent.1

Keywords

  • arithmetic formula
  • thermodynamic state
  • adiabatic change
  • inference

1. Introduction

To show that the real/physical sense of the Gödel incompleteness theorems—that the very real sense of them—is the meta-arithmetic-logical analog of the Caratheodory’s claims about the adiabatic system (that they are the analog of the sense of the II. Law of Thermodynamics), we compare the states in the state space of an adiabatic thermodynamic system with arithmetic formulas and the Peano inference is compared with the adiabatic changes within this state space. The whole set of the states now not achievable adiabatically represents the existence of the states on an adiabatic path, but this fact is not expressible adiabatically. This property of which is the analog of the sense of Gödel undecidable formula. Nevertheless, any of these states, now not achievable adiabatically in the given state space (of the given adiabatic system), is achievable adiabatically but in the redefined and wider adiabatic system with its state space divided between adiabatic and not adiabatic parts again. These states (which are achievable only when the previous subsystem is part of the new actual system, both are consistent/adiabatic) represent arithmetic but not the Peano arithmetic formulas and also are bearing the property of their whole set. Also they can be axioms of the higher/superior inference including the previous one—the general arithmetic inference is further ruled by the same and repeated principle of widening the axiomatics and with same thermodynamic analogy using the redefined and widened new adiabatic system and its settings and with the same limitation by the impossibility to proof both the consistency of the given inferential system and, in our analogy, the adiabacity of its given adiabatic analog, by means of themselves. The consistency of the inferential system and adiabacity of its analog (and their abilities generally) are defined and proved by outer construction, outer limitations, and outer settings only (compare this our claim with the Gödel’s claim for the Peano arithmetic inference “… in the Peano arithmetic system exists …”).

Caratheodory common formulation of the II. P.T.:

In our considerations, we use the states of the adiabatic system as the thermodynamic representation of the Peano arithmetically inferred formulas and the transition between the stats is then the thermodynamic model of the Peano arithmetic inference step, the consistency of the Peano arithmetics is represented by the adiabacity of the modeling thermodynamic system.

Peano Axioms/Inference Rules in the System P/Theory TPA.

♣ “1″ - arithmeticity of the Padiabacity of the L/OL.

♣ Consistent TPAinference within Pmoving along trajectories 1DLin DL/L.

♣ The states on the adiabatic trajectories, also irreversible, then model the consistently inferred/inferrable PA-FORMULAS.

Remark: Any inference within the system P2 sets the TPA-theoretical relation3 among its formulae a. This relation is given by their gradually generated special sequence aa1aqapakak+1, which is the proof of the latest inferred formula ak + 1. By this, the unique arithmetic relation between their Gödel numbers, FORMULAE x[·], x[·] = Φ(a[·]), is set up, too. The gradually arising SEQUENCE of FORMULAE x=Φais the PROOF of its latest FORMULA xk + 1.

Let us assume that the given sequence a=ao1ao2aoaqapakak+1is 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.4

Within the process of the (Gödelian) arithmetic-syntactic analysis of the latest formula ak + 1 of the proof a, we use, from the aselected, (special) subsequence aq,p,k+1of the formulae aq, ap, ak + 1. The formulae aq, ap have already been derived, or they are axioms. It is valid that q, p < k + 1, and we assume that q < p,

aq,p,k+1=aqapak+1,apaqak+1,aq,p,k+1=aqaqak+1ak+1,x=Φa=ΦΦa1Φa2ΦaqΦapΦakΦak+1=Φx=Φx1Φx2ΦxqΦxqΦxkΦxk+1lx=lΦx=lΦa=k+1,xk+1=Φak+1=lΦaGlΦa=k+1Glxxp=Φap=Φapak+1=qGlΦaΦlΦaGlΦa=qGlxImplxGlxxq=Φaq=qGlΦa=qGlx

Checking the syntactic and TPA-theoretical correctness of the analyzed chains ai, as the formulae of the system Phaving been generated by inferring (Modus Ponens) within the system P(in the theory TPA), and also the special sequence of the formulae aof the system (theory TPA), is realized by checking the arithmetic-syntactic correctness of the notation of their corresponding FORMULAE and SEQUENCE of FORMULAE, by means of the relations Form(·), FR(·), Op(·,·,·), Fl(·,·,·) “called” from (the sequence of procedures) relations Bew(·), (··)B(·), Bw(·);5 the core of the whole (Gödelian) arithmetic-syntactic analysis is the (procedure) relation of Divisibility,

FormΦai=1/0,FRΦa1i+1=1/0,oikOpxkNegxqxk+1=OpΦapΦaqΦak+1=1/0Flk+1GlxpGlxqGlx=1/0xBxk+1=1/0,Bewxk+1=1/0;Φap233GlΦaq,p,k+1&Φap71GlΦaq,p,k+1=1/0

2. Gödel theorems

Remark: The expression Sbu1u2tZxZyor the expression Sb1719tZxZyrepresents the result value of the Gödel number t[Z(x), Z(y)], which is coding the (constant) claim T (x, y) z PM has been generated by the substitution of x a y instead of the free variables X and Y in the function T (X, Y) from PM with its Gödelian code t(u1, u2) in the (arithmetized) P,

Sbu1u2tZxZy=Sb1719tZxZy

Into the VARIABLES, we substitute the SIGNS of the same type but the introduction of the term admissible substitution itself is not supposing it wordly.

- Then it is possible to work even with the expressions not grammatically correct and thus with such chains, which are not FORMULAE of the system P(and thus not belonging into the theory TPA).

Then the substitution function Sbis not possible, within the frame of the inference in the system P, be used isolately as an arbitrarily performed number manipulation—in spite of the fact that it is such number manipulation really. It is used only and just within the frame of the language LPand, above all, within the frame of the conditions specified by the právě a jenom INFERENCE of the elements of the language LTPAonly (and thus in the more limited way).

Others than/semantically (or by the type) homogenous application of the substitution function is not within the right inference/INFERENCE within the system Ppossible.6

2.1 The Gödel UNDECIDABLE CLAIM’s construction

♦ Let the Gödel numbers x and y be given. The number x is the SEQUENCE OF FORMULAE valid and y is a FORMULA of P. We define the valid constant relation Q(x, y) from the Q(X, Y) for given values x and y, X:=x, Y:= y; 17 = Φ(X), 19 = Φ(Y),78

QxyxBκSb19yZp¯BewκSb1719qZxZyqZxZy=ΦQxy,xBκy¯Bewκy=BewκyZy=BewκqZxZyE1

♦ Now we put p = 17Gen q, q = q(17, 19) q1719QXYand then,

p=17Genq1719=ΦxXQxYQXYQN0YE2

The meta-language symbol QXYor QN0Yis to be read: No xXN0is in the κ-INFERENCE relation to the variable Y (to its space of values Y).

♦ Further, with the Gödel substitution function, we put q[17, Z(p)] = r(17) = r,

rSb19qZpand thenr=Sb19q1719Zp=r17=ΦQXpE3

The Gödel number r is, by the substitution of the NUMERAL Z(p), supposedly only (by [5, 6, 7]) the CLASS SIGN with the FREE VARIABLE 17 (X); with the values p, the r contains the feature of autoreference,

r=r17=q17Zp19=q17Z17Genq1719QXp=ΦQXΦxXQxYYpQXΦQXYQXΦQN0YE4

♦ Within the Gödel number/code q, q = q [17, 19], we perform the substitution Y: = p and then X: = x and write

rZx=Sb1719q1719ZxZp=Sb17q17ZpZx=ΦQxΦQXY=ΦQxΦQN0Y=ΦQxpE5

With the great quantification of r[Z(x)] by Z(x) by the VARIABLE X (17), we have (similarly as in [4, 8]),

ZxGenrZx=17Genq17Z17Genq1719=17Genr17=17GenrΦxXΦQxΦxXQxY=ΔQXΦQXY=QN0ΦQN0YE6

2.2 Gödel theorems

I. Gödel theorem (corrected semantically by [3, 9, 10]) claims that

for every recursive and consistent CLASS OF FORMULAE κ and outside this set there is such true (“1”) CLAIM r with free VARIABLE vrrvthat neither PROPOSITION vGen r nor PROPOSITION Neg(vGen r) belongs to the set Flg(κ),

vGenrFlgκ&NegvGenrFlgκE7

FORMULA vGen r and Neg(vGen r) are not κ-PROVABLE—FORMULA vGen r is not κ-DECIDABLE. They both are elements of inconsistent (meta)system P.

II. Gödel theorem (corrected semantically according to [3, 9, 10]) claims that

if κ is an arbitrary recursive and consistent CLASS OF FORMULAE, then any CLAIM saying that CLASS κ is consistent must be constructed outside this set, and for this fact it is not κ-PROVABLE.

- Outside9 the consistent system Pκ, there is a true (“1”) formula,10 the ARITHMETIZATION of which is κ-UNPROVABLE FORMULA 17Gen r.11

♦ The fact that the recursive CLASS OF FORMULAE κ (now PAPeano Arithmetic especially) is consistent, is tested by unary relation Wid(κ), (die Widerspruchsfreiheit, Consistency) [5, 6, 7],

WidκExFormx&Bewκx¯E8

- a class of FORMULAE κ is consistent Defthere exists at least one FORMULA x [PROPOSITION x (x = 17Gen r)], which is κ-UNPROVABLE.

3. Caratheodory theorems

I. Caratheodory’s theorem (⇒) says that: ◊ If the Pfaff form has an integration factor, then there are, in the arbitrary vicinity of any arbitrarily chosen and fixed point P of the hyperplane RPRxii=1n=const., such points which, from this point P, are inaccessible along the path satisfying the equation dQ = 0.

II. Caratheodory theorem (⇐) says that: ◊ If the Pfaff form δQ=i=1nXidxi, where Xi are continuously differentiable functions of n variables (over a simply continuous area), has such a property that in the arbitrary vicinity of any arbitrarily chosen and fixed point P of the hyperplane RPRxii=1n=const., there exists such points which, from P, cannot be accessible along the path satisfying the equation dQ = 0, then this form is holonomous; it has or it is possible to find an integration factor for it.

Caratheodory formulation of the II. Law of Thermodynamics (⇔) claims that:

In the arbitrary vicinity of every state of the state space of the adiabatic system, there are such states that, from the given starting point, cannot be reached along an adiabatic path (reversibly and irreversibly), or such states which the system cannot reach at all, see the Figure 1.

Figure 1.

Adiabatic changes of the state of the system L, illustration.

Remark: Now the symbol Q denotes that heat given to the state space of the thermodynamic system from its outside and directly; QQExt; along paths l2b, l2b′, l2d, l2e, l3 is QExt = 0,QExt = 0, dQExt = 0.

The states’ θLchanges in the adiabatic system L/OL, along the trajectories lOLare expressible regularly:

Through the state space of FORMULAE of the system P, we “travel” similarly by the inference rules, Modus Ponens especially [performed by a Turing Machine TM, the inference of which is considerable as realized by the information transfer process within a Shannon Transfer Chain XKYdescribed thermodynamically by a Carnot Machine CM].

The thermodynamic model for the consistent P/TPAinference, from its axioms or formulas having been inferred so far, is created by the Carnot Machine’s activity, which models the inference. This whole Carnot Machine CM runs in the wider adiabatic system L/DLand, in fact, is, in this way, creating these states, [the TM’s, XKY‘s, configurations are then modeled by the states θiLOLof the adiabatic L/DLwith this modeling CM inside], see the Figure 2.

Figure 2.

The mutual describability of the CM, XKY and TM.

The L‘s initial imbalance starts the θLs states’ sequence on a trajectory lOLand is given by the modeled

These adiabatic trajectories lOLnow represent the norm of the consistency (and resultativity) of the P/TPA-inference/computing process expressible also in terms of the information transfer/heat energy transformation.

The adiabatic property of the thermodynamic system Lis always created over the given scales of its state quantities—over their scale for a certain “creating” original (and not adiabatic) system T, and by its outerly specification or the design/construction by means of heat/adiabatic isolation of the space Vmax of the original system Tthat the system (L/T) can occupy, and after the system Lhas been (as the adiabatic isolated original system T) designed and set in the starting state θ1, see Figure 1. The state θ4 is a stateof the set of states {◊}. These states are those ones in the Figure 1, which, although they are in the given scale of state quantities U and V of the state space OLof the system Lconsidered, UUminUmaxand VVminVmax, are within it [in (the state space OLof) L] by permitted (adiabatic, dQExt = 0) changes l2b, l2b′, l2d, l2e and l3, inaccessible. And certainly, thermodynamic states □ beyond these scales, within the hierarchically higher systems, are not accessible from the inside of the system L/Titself, without its (not adiabatical) widening, either, see the Figure 1.

. Without violation of the adiabacity of the system L, it is not possible to reach the state θ4 from the state θ1 along any simple path l2b, l2b′, l2d, l2e in the state space OL,

♣ However, outside the adiabacity of the system Lexpressed by the relation dQExt = 0, which means under the opposite requirement dQExt0, it is possible to design or to construct a (nonadiabatic) path linking a certain point/state of the state space OLlocated, e.g., on l2e with the point/state θ4; for example, it is the path l4 from θ1 to θ4, now in a certain nonadiabatic system N, NTwhere, from the view of possibilities of changes of the state, see Figure 1, is valid that

OL/T,OL/T;l2bl2bl2dl2el3,OL/TOL=ON=OTOL/TTNLE9

. Further, it is possible to create for this nonadiabatic system Nan alternative adiabatic system LDLDLenabling adiabatic-isochoric changes, e.g., θ2e → θ4.

.. Both the new adiabatic system Land its nonadiabatic “model” Ncan be a subsystem of another but also adiabatic and imminently superior system L+having another/wider range of the state quantities than it was for the original systems Land N, OLOL/NOL+. Then the path l4 in the state space OL/Nof the system L/Nwill be, from the point of Lof the imminently superior adiabatic system L+, the adiabatic one—the system Lis already isolated in L+and the system L+itself is already created in a certain system Limminently superior to it, as an isolated/adiabatic substitute for the system NOL/NOL+OL+/NOL.

.. From the view of the possibilities to change the state, or from the view of the energetic relations (), it is possible, see the Figure 1, to write,

LLL+LL,NNN+NNE10

NL+E, Nis implemented in L, N+LE, Nis implemented in L+N+LE,N+is implemented in v L, …

We introduce a symbol lOL·for adiabatic paths in the state spaces OL·,

lQLl2bl2bl2dl2el3,lQLl2bl2bl2dl2el3l2elθ2e,θ4l5,lQL+,lQL,lQLlQLlQL+lQLE11

The states from the sets OLlQL,OL'lQL,QL+lQL,OLlQL,in the view of adiabacity and specification of the system Lare forming, within the hierarchy of the systems L,L,L+,L,,a certain set OL=, which is in the framework of the system Linaccessible/unachievable as a whole and also in any of its subset and member. However, the L-inaccessibility (adiabatic inaccessibility, especially of {◊} in the state space OL/T) also means existence of the paths lQLof the adiabatic system L. In the sense of the domain of solution of its (the L‘s) state equations, they cannot be part of the functionality of L(but mark it).

4. Analogy between adiabacity and PA-inference

♣ Now the states on the adiabatic paths lQL(of changes of the state of the adiabatic system L) are considered to be the analogues of PA-arithmetic claims/claims of the Peano Arithmetic theory TPA(formulated/inferred/proved in P),

- adiabacity of the system Lis the analog of consistency of the system Pκand

. the set lQLof adiabatic paths in OL/Tis an analog of PA-theory TPA; then, adiabatic analogy of the higher consistent inferential system Pis by L, PP,.

.. Then the given specific adiabatic path l2b, l2b′, l2d, l2e, l3 is an analog of certain deducible thread xBxkof the claim xk of the theory TPA, where

xBxk=x1x2xk1xkBxk=1x1AXIOMSPandx1θ1x1,x2,,xk1,xkTPAandxkθθ2bθ2bθ2dθ2eθ3x2,,xk1θ{{l2bθ2b,l2bθ2b,l2dθ2d,l2eθ2e,l3θ3}θ1}E12

♣ The states from the space QL/Tof the system L/Tsatisfying the range of values of the state quantities ppminpmax, VVminVmax, TTminTmax/UUminUmax), which are inaccessible along any of the adiabatic paths from lQL, that means they are the states ◊ from the difference OL/TlQL, shortly said from TL, are considered to be analogues of not PA-claims such as, e.g., the Fermat’s Last Theorem.12 So, they are analogues of all-the-time true (“1”) arithmetic but not-PA-arithmetic claims. From the point of adiabacity of the system L, they (◊) are only some thermodynamic states of its “creating” system T, and they are from the common range of values of the state quantities for Tand L. From the point of expressing possibilities it as always true

LNTTQL/TE13

[Symbol Tdenotes thermodynamic theory as a whole and symbol QL/Tis a mark for a transitive and reflexive closure of the set of (any) claims about systems L//T.].

The whole set OLof states inaccessible in a given scale of state quantities of the system L/Talong the arbitrary adiabatic path from lQLin the system L(states ◊), as well as the set of L-inaccessible statesoutside this scale, see Figure 1, are considered now to be the thermodynamic bearer of analogy of the semantics of the Gödel’s UNDECIDABLE PROPOSITION 17Gen r,

17GenrQL=OL=QLlQL,QLQLQL/TE14

- The states from OL(from QLlQL,QLlQL,QLlQL+,,QLlQL,,QLlQL,) inaccessible by permitted changes in currently used systems L,L,L+,L,(within the scale of values of their state quantities and also out of this scale) confirm both existence and properties of these systems L,L,L+,L,; they confirm adiabacity of changes lQL,lQL,lQL+,lQL,running in them.

For (to illustrate our analogy) a supposedly countable set of states along the paths lQLof changes of the state of the system L(for simplicity we can consider the isentrop l2e only), the PROPOSITION 17Gen r is a claim of countability set nature, the analog OLof which is formulated in the set QL/T; it as valid that

QL/TQLandQLlQLQLQL/TQLlQLQLandQLlQLlQLE15

4.1 Analogy between Caratheodory and Gödel theorems

We claim that, II. Caratheodory theorem,

if an arbitrary Pfaff form δQExt=i=1nXidxi, where Xi are functions of n variables, continuously differentiable (over a simply continuous domain) has such a quality that in the arbitrary vicinity of arbitrarily chosen fixed point P of the hyperplane R[PR,Rxii=1n=C=const.] there exists a set of points inaccessible from the point P along the path satisfying the equation dQExt = 0, then it is possible to find an integration factor for it and then this form is holonomous. In a physical sense and, by means of the Thermodynamics language,

lQLQLQLQLQL/TE16

it says what, in its consequence [w 17Gen r, (8)] and in a meta-arithmetic-logical way, the II. Gödel theorem (corrected semantically by [3, 9, 10]) claims;

♣ if κ is an arbitrary recursive and consistent CLASS OF FORMULAE, then any CLAIM (written as the SENTENCIAL and as such, representing a countable set of claims, which are its implementations) saying that CLASS κ is consistent must be constructed outside this set and for this fact it is not κ-PROVABLE/is κ-UNPROVABLE or cannot be κ-PROVABLE. In fact, it is a part of the inconsistent metasystem P.

- Outside the consistent system Pκ, there is a true (“1”) formula whose ARITHMETIZATION is κ-UNPROVABLE FORMULA/PROPOSITION/CLAIM or code 17Gen r”.13

. In a physical sense and by the Thermodynamics language,

QL/TQLQLQLlQLE17

♣ It is possible to claim that, I. Caratheodory theorem,

if an arbitrary Pfaff form δQExt=i=1nXidxihas a integration factor, then there are in the arbitrary vicinity of an arbitrarily chosen fixed point P of the hyperplane Rsome points inaccessible from this point PPRxii=1n=const.along the path satisfying the equation dQExt = 0. In a physical sense and by means of the Thermodynamics language,

QLlQLQL/TQLlQLE18

it says what, in a meta-arithmetic-logical way, the I. Gödel theorem (corrected semantically by [3, 9, 10]) claims;

♣ for every recursive and consistent CLASS OF FORMULAE κ and outside this set, there is such a true (“1”) CLAIM r with free VARIABLE vrrvthat neither PROPOSITION vGen r nor PROPOSITION Neg(vGen r) belongs to the set Flg(κ),

vGenrFlgκ&NegvGenrFlgκE19

FORMULA vGen r and Neg(vGen r) are not κ-PROVABLE—FORMULA vGen r i s not κ-DECIDABLE. They are elements of inconsistent (meta)system P.

For us, as an isolated system L, to achieve such a “state,” it is necessary to consider the states with values of state quantities which are not a part of the domain of solution of the state equation for L. The system Lhas not been designed for them (so, we are facing inconsistency). For example, the required volume V and temperature T should be greater than their maxima Vmax and Tmax achievable by the system L. In order “to achieve” them, the system Litself would have to “get out of itself,” and in order to obtain values V and T greater than Vmax and Tmax, it would have to “redesign”/reconstruct itself. However, it is us, being in a position of the hierarchically higher object, who has to do so, from the outside the state space QL/T(from the outside the volume Vmax), which the system may occupy now.14

- This “procedure” corresponds to the CLAIM/PROPOSITION/FORMULA 17Gen r construction by means of (Cantor’s) diagonal argument and Caratheodory proof.

♣ The states unachievable within the state spaces of the systems L,L,L+,L,or inaccessible from them are creating, as a whole, a certain class of equivalence or macrostate OLQLQLlQLin hierarchy of the state spaces, from the point of their possible development, of always superior systems—LL+Lfor L,L+Lfor L,Lfor L+, . The existence of the macrostate OL, already beginning from the original system L(macrostate OL), confirms the existence of the currently considered (adiabatic) system Land its properties, especially its adiabacity. And by this, in our analogy, it also con rms the consistency of its arithmetic/mathematical analog P,P,P+,(a complement of the set cannot exist without this set) and, on the contrary,

QL/TQL,QL/TQL;QLL&LQLQLLE20

- Based just upon this point of view, we assign the set/macrostate or equivalence class OLthe meaning of the bearer of the sense of the Gödel’s UNDECIDABLE PROPOSITION 17Gen r for P,

- the L-unachievability of the set OLis in the position of the analog for this, in fact, methodological axiom which has been formulated in a certain hierarchically higher inferential (meta)system P,POL. In accordance with the above and with Figure 1, we write for L/P

θ4QLlQL==QLQLlQLQLE21
lQLθ4lQLQLlQLQLlQLlQLQLlQLlQLQLlQLlQLlQLQLlQLQLlQLlQLlQLlQLQLlQLQLlQL,

and further, for the theory lOL/TPA, following (1)(6) and [4], we write

lQLTPA,cardlQL=cardTPA=0cardQL=1,cardQLlQL=1lQL17,1919QLlQLy=q1719θ,p=17Genq1719yZy=q17yθθθθθθθθθθ,E22

For 19: = Z(p) is p[Z(p)] = r(17) and r17and so we can write neatly

θlQL17Gen,θlQLθ17Genq171917GenrθlQLθθlQLθ17GenrlQLlQLOLlQL,E23

which is the same as (21).

- It is obvious from our thermodynamic analogy that CLAIM/PROPOSITION 17Gen r for has to be true and in connection with Gödel’s II. theorem, and in accordance with Caratheodory we claim that

QL17GenrforL/PvGenrforL/PL+E24

The notation 17Gen r itself expresses the property of the system Pand also the theory TPA, just as an subject which itself is not and cannot be the object of its own, and thus its notation is not and cannot be one of the objects of the system P[similarly, as (17) is valid, OLQLlQL].

Demonstration: Following (8) WidP17Genr, we claim for the systems L/Pthat

dQExtL=0w,dQExtL=0L;wdQExtL=0,LdQExtL=0LOL,QL17Genr;OLL,L17GenrLOLw17Genr;OLL17Genrwso,thatOLL17Genrw&LOLw17Genrand thenOL17GenrE25

I. Gödel theorem (corrected semantically by [3, 9, 10]):

For every recursive and consistent CLASS OF FORMULAE κ, and outside this set, there exists the true (“1”) CLAIM r with a free VARIABLE v that neither the CLAIM vGen r nor the CLAIM Neg(vGen r) belongs to the set Flg(κ)

vGenrFlgκ&NegvGenr/Flgκ,

CLAIMS vGen r and Neg(vGen r) are not κ-PROVABLE, the CLAIM vGen r is not κ-DECIDABLE.

[They are elements of the formulating/syntactic metasystem κ, inconsistent against κ].

II. Gödel theorem (corrected semantically by [3, 9, 10]):

If κ is an arbitrary recursive and consistent CLASS OF FORMULAE, then any CLAIM saying that CLASS κ is consistent must be constructed outside this set and for this fact, it is not κ-PROVABLE.

The consistency of the CLASS OF FORMULAE κ is tested by the relation Wid(κ).

The FORMULAE class κ is consistent.

at least one κ-UNPROVABLE CLAIM x exists.

Now x = 17Gen r P/TPA, κ=TPA, TPAPP

Then, semantically understood and with the language of logic and meta-arithmetics, the full meaning of the Gödel proof expresses the universal validity of the II. Law of Thermodynamics.15

5. 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 Gödel formula and its negation are not provable and decidable within it. They are constructed, not inferred, by the 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 Gödel 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 Gödel 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 Gödel 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 Gödel 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 (in fact, of the properties of the information transfer).

Any description of real objects, no matter how precise, is only a model of them, of their properties and relations, making them available in a specified and somewhat limited (compared with the reality) point of view determined by the description/model designer. This determination is expressed in definitions and axiomatics of this description/model/theory—both with definitions and by axioms and their number. Hence, realistically/empirically or rationally, it will also be true about (objects of) reality what such a model, called recursive and able-of-axiomatization, does not include. With regard of reality any such a model is axiomatically incomplete, even if the system of axioms is complete. In addition, and more importantly, this description/model of objects, of their properties and possible relations (the theory about reality) cannot include a description of itself just as the object of reality defined by itself (any such theory/object is not a subject of a direct description of itself). The description/model or the theory about reality is a grammar construction with substitutes and axiomatization and, as such, it is incomplete in the Gödelian waythe grammar itself does not prevent a semantical mixing; but any observed real object cannot be the subject of observation of itself and this is valid for the considered theory, just as for the object of reality, too. No description of reality arranged from its inside or created within the theory of this reality can capture the reality completely in wholeness of its all own properties. It is impossible for the models/theories considered, independently on their axiomatization. They are limited in principle [in the real sense of the Gödel theorems (in the Gödelian way)].

Now, with our better comprehension, we can claim that the consistency of the recursive and axiomatizable system can never be proved in it itself, even if the system is consistent really. The reason is that a claim of the consistency of such a system is designable only if the system is the object of outer observation/measuring/studies, which is not possible within the system itself. Ignoring this approach is also the reason for the formulation of the Gibbs paradox and Halting Problem. Also, our awareness of this fact results in our full understanding of the meaning and proof of the Gödel theorems, very often explained and described incomprehensibly, even inconsistently or paradoxically, and which is parallel with the way of the Caratheodory proof of the II. Thermodynamic Principle.

♣♦◊

Acknowledgments

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

Many thanks are to be expressed to my brother Ing. Petr Hejna for his help with English language and formulations of both this and all the previous texts.

A. Appendix

A.1 Summarizing comparison

Under the adiabacity, [d]QExt = 0, of the system L, it is not possible to derive such a CLAIM that is stating this adiabatic supposition. This CLAIM is constructible not adiabatically, outside the adiabatic Lonly.

Under the consistency of the system P, it is not possible to derive such a CLAIM that is stating this consistency supposition. This CLAIM is constructible purely syntactically, outside the consistent Ponly (in PP) (Figure A1).

Figure A1.

Example of not distinguishing the reality and its image.

Without Pwe could not know that P is not self-referencing and is consistent.

A.2 The proof way of Caratheodory theorems

I. Let the form δQ=i=1nXixihas the integration factor v and let dR=i=1n1vXidxi. Then the Pfaff equation δQ=i=1nXidxi=0has the solution in the form Rx1xk=const. and this solution represents a family of hyperplanes in n-dimensional space, not intersecting each other. Let us pick now the point Px10xn0determined by our choice of const. = C. Only the points lying in the hyperplane Rx10xn0are accessible from the point P along the path satisfying the condition dQ = 0. All the points not lying in this hyperplane are inaccessible from the point P along the path satisfying the condition dQ = 0 (Figure A2).

Figure A2.

The proof way of the Caratheodory theorems.

II. Let us pick the point V, e.g., from R3, lying in a vicinity of the point P, which is not accessible from P following the path dQ = 0. Let g be a line going through the point P and let g be oriented (g) in such way that it does not satisfy the condition dQ = 0. The point V and the line g determine a plane Xi = Xi(u, v), i = 1, 2, 3. Let us consider a curve k in this plane, going through the point V (u0, v0) in that way (g) that dQ = 0 is supposedly valid along this curve. There is only one curve k for the point V (u0, v0). It lies in our plane, the plane Xi = Xi(u, v), and then it is valid for it dXi=Xiudu+Xivdvand, considering dQ = 0 along k, we get i=13XiXiudu+i=13XiXivdv=0.

The curve k, however, intersects the line g in the point R, which is inaccessible from the point P along the path with dQ = 0 (for dQRg0). Otherwise, the point V would also be accessible from the point P through R and kdQRk=0, which is a conflict with the original assumption. By a suitable selection of V, it is possible to have the point R arbitrarily close to the point P; in the arbitrary vicinity of the point P, there are points inaccessible from the point P along the path with dQ = 0. Now, let us pick a line g′ parallel to the line g, and a cylinder C going through these two lines. We consider that the curve k satisfying the relation dQ = 0 is on this cylinder C′ goes through the point P and intersects the line g′ in the point M.

Now, let us consider another cylinder C′ as the continuation of C with g′ and g. Let us use the symbol k′ for the continuation of the curve k in C′. Then the curve k′ must intersect the line g in the point P. Otherwise, it would be possible to deform the plane C′ as much as to get C, thus continually merging the intersecting point N into the point P and at the moments of discrepancy of the points P and N, it would be possible to reach the point P from the point N along the line g (supposedly with dQ = 0). However, the condition dQ = 0 is not valid there (dQRg0). By deforming C′ into C, the k and k′ would close a plane F where dQ = 0. If the equation of this plane has the form Rxii=13=const., then the equation dQ = 0 has a solution—an integration factor for the Pfaff form δQ=i=13Xidxiexists [11].

A.3 Information thermodynamic concept removing autoreference

The concept for ceasing the autoreference, based on the two Carnot Cycles disconnected as for their heaters and described informationally, shows the following Figure A3. (also see [1, 2, 4]):

Figure A3.

The concept for ceasing the autoreference.

For ∆A″, it is valid in the cycle Othat

ΔA=ΔQW1T0TW=ΔQW.TWTW1T0TW==ΔQWTWTWT0TW=kHXTWT0=kHXTW1T0TW=kHXTW1β=kTWHYE26

and, further, for ∆A in the cycle O, we have

ΔA=kHXTW1β=kHXTW1T0TWE27

and thus, for the cycles Oand O, it is valid that

ΔAkTW=HX1T0TW=HX1β=HXηmaxΔAkTW=HX1T0TW=HX1β=HXηmaxE28

For the whole work ∆A* of the combined cycle OO, we have

ΔA=ΔAΔA=kTWHX1βkTWHX1β>0E29

Then, for the whole change of the thermodynamic entropy within the combined cycle OO(measured in information units Hartley, nat, bit) and thus for the change of the whole information entropy H*(Y*), it is valid that

HY=ΔAkTW=HX1βTWTW1β=HX1T0TWTWTW+T0TW=HX1TWTWE30

It is valid, for ∆A* is a residuum work after the work ∆A has been performed at the temperature TW. Evidently, the sense of the symbol TW(within the double cycle OOand when ∆Q0 = ∆Q″0) is expressible by the symbol T* 0, which is possible, for the working temperatures of the whole cycle OOare TW and T″W = T*0. The relation (30) expresses that fact that the double cycle OOis the direct Carnot Cycle just with its working temperatures TW > T″W = T*0. In the double cycle OO, it is valid that

β=ΔQ0ΔQW=ΔQ0TWΔQWTW=HYXHY=T0TW,TW=T0,cyklusOβ=ΔQ0ΔQW=ΔQ0TWΔQWTW=HXYHX=T0TW,cyklusOββ=TWTW=T0TWβE31

and then, by (30) and (31) is writable that

ΔAkTW=HX1β=HX1HXYHYHYXHX>0E32

It is ensured by the propositions TW > T″W, T″0 = T0 and also by that fact that the loss entropy H(X|Y) is described and given by the heat ∆Q0 = ∆Q0. But in our combined cycle OO, it is valid too that

HX=ΔQWkTW=ΔQWkTW=HY=ΔQWkT0E33

and we have

HXYHYX=β<1E34

For the whole information entropy ΔAkTW(the whole thermodynamic entropy SCin information units) and by following the previous relations also it is valid that

ΔAkTW=HYHYβ=HY1T0TW=HY1HXYHXYE35

And thus, the structure of the information transfer channel K[expressed by the quantity H(X|Y)] is measurable by the value H*(Y*) from (32) and (35). Symbolically, we can write, using a certain growing function f,

HY=ΔAkTWfHXY>0E36

The cycles O,O, and OOare the Carnot Cycles, and thus from their definition and construction, they are imaginatively16 in principle, the infinite cycles; in each of them the following criterion of an infinite cycle (see [12]) it is valid inevitably,

TXY=HXHXY=HY>0andΔSL=0E37

The construction of the cycle OOenables us to recognize that the infinite cycle Ois running. In our case, it is the infinite cycle from (5), (6) and also from [4, 8, 10],

QXY,QXΦQXY,QXΦQXΦQXY,QN0Y,QN0ΦQN0Y,QN0ΦQN0ΦQN0Y,E38

Notes

  • The reader of the paper should be familiar with the Gödel proof’s way and terminology; SMALL CAPITALS in the whole text mean the Gödel numbers and working with them. This chapter is based, mainly, on the [1, 2, 3, 4]. This paper is the continuation of the lecture Gödel Proof, Information Transfer and Thermodynamics [4].
  • Formal arithmetic inferential system.
  • Peano Arithmetics Theory.
  • For simplicity. The ‘real’ inference is applied to the formula ai + 1 for i = o.
  • Formula, Reihe von Formeln, Operation, Folge, Glied, Beweis, Beweis, see Definition 1–46 in [5, 6, 7] and by means of all other, by them ‘called’, relations and functions (by their procedures).
  • Substitution function Sb⋯⋅⋯ is, in this way, similar to the computer machine instruction which itself, is always able to realize its operation with its operands on the arbitrary storage place, but practically it is always applicated within the limited address space and within the given operation regime/mode of the computer’s activity only (e.g. regime/mode Supervisor or User).
  • Φ and Z represents the Gödel numbering and Sb the Substitution, B, Bew the PA-arithmetic Proof.
  • Following the Gödel Proposition V (the first part) [5, 6, 7].
  • Far from (!) “In ….” in [5, 6, 7]
  • Far from “… [PA-]arithmetic and sentencial/SENTENCIAL” in [5, 6, 7].
  • Any attempt to prove/TO PROVE it (to infer/to TO INFER it) in the system Pκ assumes or leads to the requirement for inconsistency of the consistent (!) system Pκ (in fact we are entering into the inconsistent metasystem P∗ - see the real sense [4, 9] of the Proposition V in [5, 6, 7]).
  • Alternatively Goldbach’s conjecture.
  • Any attempt to prove/TO PROVE it (to infer/TO INFER it) within the system Pk assumes or leads to the Circulus Viciosus.
  • This also involves introduction of the representative θ0 of Fermat’s Last Theorem provided we are speaking about L with lOL and provided we require enlargement L′ in order to get L′≅P′. The specific states accessible in the state space OL={p∈pminpmax, V∈VminVmax, T∈TminTmax/U∈UminUmax,… of the isolated system L through reversible or irreversible changes other than adiabatic are thermodynamic analogy (interpretation) of the enlargement of the axiomatics of the original system Pκ to the new system P′,P+,…, similar/relative to the Pκ. Such an enlargement of the system P to a certain system P⋅⋅ enabled Andrew Wiles to prove the Fermat’s Last Theorem. Through its representative θ0 we enlarge L to L′,L′≅P′.
  • Our consideration is based on the similarity between the Cantor diagonal argument used in construction of the Gödel Undecidable Formula and the proof way of the Caratheodory theorems; adiabacity/consistency is prooved by leaving them and sustaining their validity - paradox.
  • When an infinite reserve of energy would exist.

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Bohdan Hejna (October 30th 2019). Common Gnoseological Meaning of Gödel and Caratheodory Theorems, Ontological Analyses in Science, Technology and Informatics, Andino Maseleno and Marini Othman, IntechOpen, DOI: 10.5772/intechopen.87975. Available from:

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