**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 metaarithmetics, 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

<sup>1</sup> 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 [1–4]. This paper is the continuation of the lecture *Gödel Proof, Information Transfer and Thermodynamics* [4].

*Ontological Analyses in Science,Technology and Informatics*

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 … ").

**Peano Axioms/Inference Rules in the System** P**/Theory** T PA**.**

*Common Gnoseological Meaning of Gödel and Caratheodory Theorems*

*DOI: http://dx.doi.org/10.5772/intechopen.87975*

♣ "1″ - arithmeticity of the P ffi adiabacity of the <sup>L</sup>*=*OL.

consistently inferred/inferrable *PA-FORMULAS*.

*sequence a*! *<sup>a</sup>*1, … , *aq*, … , *ap*, … , *ak*, *ak*þ<sup>1</sup>

Let us assume that the given sequence *a*

*FORMULAE x* ¼ Φ *a*

formula *ak* + 1 of the proof *a*

*aq*,*p*,*k*þ<sup>1</sup> ����! <sup>¼</sup> *aq*, *ap*, *ak*þ<sup>1</sup>

*x* ¼ Φ *a*

¼ Φ *x*

*l x*ð Þ¼ *l* Φ *x*

*xp* ¼ Φ *ap*

*xq* ¼ Φ *aq*

<sup>2</sup> Formal arithmetic inferential system.

<sup>3</sup> Peano Arithmetics Theory.

**3**

♣ Consistent <sup>T</sup> PA inference within P ffi moving along trajectories 1DL in DL*=*L. ♣ The states on the adiabatic trajectories, also irreversible, then model the

**Remark**: Any *inference* within the system <sup>P</sup><sup>2</sup> sets the <sup>T</sup> PA-*theoretical* relation<sup>3</sup> among its formulae *a*½ �� . This relation is given by their gradually generated *special*

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*

is a special one, and that, except of axioms (axiomatic schemes) *a*01, *…* , *ao*, it has

Within the process of the *(Gödelian) arithmetic-syntactic analysis* of the latest

*aq*,*p*,*k*þ<sup>1</sup> ����! of the formulae *aq, ap, ak* + 1. The formulae *aq*, *ap* have already been derived,

� �, *ap* ffi *aq* <sup>⊃</sup>*ak*þ1, *aq*,*p*,*k*þ<sup>1</sup> ����! <sup>¼</sup> *aq*, *aq* <sup>⊃</sup>*ak*þ1, *ak*þ<sup>1</sup>

� �, … , Φ *ap*

� � ∗ … ∗ Φ *xq*

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

!� � <sup>¼</sup> ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> *Gl x*

� � � �

been generated by the correct application of the rule *Modus Ponens only*.

!, we use, from the *a*

or they are axioms. It is valid that *q, p < k* + 1, and we assume that *q < p*,

!� � <sup>¼</sup> Φ Φð Þ *<sup>a</sup>*<sup>1</sup> , <sup>Φ</sup>ð Þ *<sup>a</sup>*<sup>2</sup> , … , <sup>Φ</sup> *aq*

!� � <sup>¼</sup> <sup>Φ</sup>ð Þ *<sup>x</sup>*<sup>1</sup> <sup>∗</sup> <sup>Φ</sup>ð Þ *<sup>x</sup>*<sup>2</sup> <sup>∗</sup> … <sup>∗</sup> <sup>Φ</sup> *xq*

! h i � � <sup>¼</sup> *<sup>k</sup>* <sup>þ</sup> 1,

� � <sup>¼</sup> *qGl*<sup>Φ</sup> *<sup>a</sup>*

!� � <sup>¼</sup> *qGl x*

<sup>4</sup> For simplicity. The 'real' inference is applied to the formula *ai* + 1 for *i* = *o*.

! h i � � *Gl* <sup>Φ</sup> *<sup>a</sup>*

! h i � � <sup>¼</sup> *<sup>l</sup>* <sup>Φ</sup> *<sup>a</sup>*

� � <sup>¼</sup> <sup>Φ</sup> *ap* <sup>⊃</sup>*ak*þ<sup>1</sup>

¼ *qGl xImp l x* ½ � ð Þ *Gl x*

� � <sup>¼</sup> *qGl*<sup>Φ</sup> *<sup>a</sup>*

*xk*þ<sup>1</sup> ¼ Φð Þ¼ *ak*þ<sup>1</sup> *l* Φ *a*

� �, which is the *proof* of the latest inferred

!� � is the *PROOF* of its latest *FORMULA xk* + 1.

! <sup>¼</sup> *ao*1, *ao*2, … , *ao*, … , *aq*, … , *ap*, … , *ak*, *ak*þ<sup>1</sup> � �

!*selected*, (special) subsequence

� �,

� �, … , <sup>Φ</sup>ð Þ *ak* , <sup>Φ</sup>ð Þ *ak*þ<sup>1</sup>

! h i � � *Gl*<sup>Φ</sup> *<sup>a</sup>*

� � <sup>∗</sup> … <sup>∗</sup> <sup>Φ</sup>ð Þ *xk* <sup>∗</sup> <sup>Φ</sup>ð Þ *xk*þ<sup>1</sup>

!� �

4

### **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.
