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


**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 sequence a*! *<sup>a</sup>*1, … , *aq*, … , *ap*, … , *ak*, *ak*þ<sup>1</sup> � �, 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* ¼ Φ *a* !� � is the *PROOF* of its latest *FORMULA xk* + 1.

Let us assume that the given sequence *a* ! <sup>¼</sup> *ao*1, *ao*2, … , *ao*, … , *aq*, … , *ap*, … , *ak*, *ak*þ<sup>1</sup> � � is a special one, and that, except of axioms (axiomatic schemes) *a*01, *…* , *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 *a* !*selected*, (special) subsequence *aq*,*p*,*k*þ<sup>1</sup> ����! of 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*,

$$\overline{a\_{q,p,k+1}} = \left[a\_q, a\_p, a\_{k+1}\right], \quad a\_p \cong a\_q \supset a\_{k+1}, \overline{a\_{q,p,k+1}} = \left[a\_q, \ a\_q \supset a\_{k+1}, a\_{k+1}\right],$$

$$\mathbf{x} = \Phi\left(\overline{a}\right) = \Phi\left(\left[\Phi(a\_1), \Phi(a\_2), \dots, \Phi(a\_q), \dots, \Phi(a\_p), \dots, \Phi(a\_p), \dots, \Phi(a\_k), \Phi(a\_{k+1})\right]\right)$$

$$= \Phi\left(\overline{\mathbf{x}}\right) = \Phi(\mathbf{x}\_1) \* \Phi(\mathbf{x}\_2) \* \dots \* \Phi(\mathbf{x}\_q) \* \dots \* \Phi(\mathbf{x}\_q) \* \dots \* \Phi(\mathbf{x}\_k) \* \Phi(\mathbf{x}\_{k+1})$$

$$I(\mathbf{x}) = I\left[\Phi(\overline{\mathbf{x}}^\cdot)\right] = I\left[\Phi\left(\overline{\mathbf{a}}^\cdot\right)\right] = k+1,$$

$$\mathbf{x}\_{k+1} = \Phi(a\_{k+1}) = I\left[\Phi\left(\overline{\mathbf{a}}^\cdot\right)\right] Gl\left(\overline{\mathbf{a}}\right) = (k+1)Gl \,\mathbf{x}$$

$$\mathbf{x}\_p = \Phi(a\_p) = \Phi(a\_p \supset a\_{k+1}) = qGl\Phi\left(\overline{\mathbf{a}}\right) \* \Phi(\supset) \* l\left[\Phi\left(\overline{\mathbf{a}}^\cdot\right)\right] Gl\left(\overline{\mathbf{a}}\right)$$

$$= qGl \,\mathbf{x} lmp \, l(\mathbf{x}) \,\overline{\mathbf{G}} \,\mathbf{x}$$

$$\mathbf{x}\_q = \Phi(a\_q) = qGl\Phi\left(\overline{\mathbf{a}}\right) = qGl \,\mathbf{x}$$

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

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

<sup>4</sup> For simplicity. The 'real' inference is applied to the formula *ai* + 1 for *i* = *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 (theory <sup>T</sup> PA), is realized by checking the *arithmeticsyntactic* 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*(�);<sup>5</sup> the core of the whole (Gödelian) arithmetic-syntactic analysis is the (procedure) relation of *Divisibility*,

*INFERENCE* **of the elements of the language** LT PA only (and thus in the more

**2.1 The Gödel** *UNDECIDABLE CLAIM***'s construction**

*Common Gnoseological Meaning of Gödel and Caratheodory Theorems*

*Q x*ð Þ� , *y xB<sup>κ</sup> Sb*

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

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

♦ Let the Gödel numbers *<sup>x</sup>* and *<sup>y</sup>* be given. The number *<sup>x</sup>* is the *SEQUENCE OF FORMULAE* valid and *y* is a *FORMULA* of P. We define the valid constant relation

<sup>5</sup> � *Bew<sup>κ</sup> Sb*

*p* ¼ 17*Gen q*ð Þ¼ 17, 19 Φ½ � ∀*<sup>x</sup>*<sup>∈</sup> j*Q x*ð Þ , *Y* ≜ *Q*ð Þ , *Y* ≜ *Q*ð Þ 0, *Y* (2)

The meta-language symbol *Q*ð Þ , *Y* or *Q*ð Þ 0, *Y* is to be read: **No** *x*∈ ð Þ<sup>0</sup> **is in**

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

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

♦ Within the Gödel number/code *<sup>q</sup>*, *<sup>q</sup>* <sup>=</sup> *<sup>q</sup>* [17, 19], we perform the substitution

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

<sup>7</sup> Φ and *Z* represents the *Gödel numbering* and *Sb* the *Substitution*, *B*, *Bew* the *PA*-arithmetic *Proof*.

*q*ð Þ 17, 19

19

1

*Z p*ð Þ

17 19

1

(1)

(4)

CCA

*Z x*ð Þ *Z y*ð Þ

CA <sup>¼</sup> *<sup>r</sup>*ð Þ¼ <sup>17</sup> <sup>Φ</sup>½ � *Q X*ð Þ , *<sup>p</sup>* (3)

≜ *Q X*½ � , Φ½ � *Q*ð Þ , *Y* ≜ *Q X*½ � , Φ½ � *Q*ð Þ 0, *Y*

is, in this way, similar to the computer *machine instruction* which itself,

*q*

0

BB@

*Q*(*x, y*) from the *Q*(*X, Y*) for given values *x* and *y*, *X*:=*x*, *Y*:= *y*; 17 = Φ(*X*),

1

3 7 7

*qZ x* ½ �¼ ð Þ, *Z y*ð Þ Φ½ � *Q x*ð Þ , *y* , *xB<sup>κ</sup> y*<sup>0</sup> � *Bew<sup>κ</sup> y*<sup>0</sup> ð Þ¼ *Bewκ*½ �¼ *yZ y* ½ � ð Þ *Bewκ*½ � *qZ x* ½ � ð Þ, *Z y*ð Þ

CCA

♦ Now we put *<sup>p</sup>* = 17*Gen q*, *<sup>q</sup>* <sup>=</sup> *<sup>q</sup>*(17, 19) *<sup>q</sup>*ð Þ 17, 19 <sup>≜</sup> *Q X*ð Þ , *<sup>Y</sup>* � � and then,

**the** *κ*-*INFERENCE* **relation to the variable** *Y* (to its space of values ).

0

B@

*r* ¼ *r*ð Þ¼ 17 *q*½17, *Z p*½ � ð Þ 19 � ¼ *q*½ � 17, *Z*½ � 17*Gen q*ð Þ 17, 19 ≜ *Q X*ð Þ , *p*

19

*Z p*ð Þ

*y*

0

BB@

limited way).

<sup>P</sup> possible.<sup>6</sup>

19 = Φ(*Y*),7,8

*r*≔*Sb*

0

B@

19 *q*

1

CA and then*<sup>r</sup>* <sup>¼</sup> *Sb*

*p,* the *r* contains the feature of *autoreference*,

¼ Φ½ � *Q X*½ � , Φ½ � ∀*<sup>x</sup>*∈*<sup>X</sup>*j*Q x*ð Þ , *Y <sup>Y</sup>*≔*<sup>p</sup>*

� ⋯ !

computer's activity only (e.g. regime/mode *Supervisor* or *User*).

<sup>8</sup> Following the Gödel Proposition *V* (the first part) [5–7].

*Y*: = *p* and then *X*: = *x* and write

<sup>6</sup> *Substitution function Sb* <sup>⋯</sup>

**5**

*Z p*ð Þ

$$\begin{split} &\text{Form}[\Phi(a\_{i})] = \ & \text{"1"} / \ \text{"0"}, \quad \text{FR}\left[\Phi\left(\overline{a\_{1}^{i+1}}\right)\right] = \ & \text{"1"} / \ \text{"0"}, \ \ o \le i \le k \\ & \text{\$Op\$}\left[\mathbf{x}\_{k}, \text{Neg}\left(\mathbf{x}\_{q}\right), \ \mathbf{x}\_{k+1}\right] = \text{Op}\left[\Phi(a\_{p}), \ \Phi\left[\sim\left(a\_{q}\right)\right], \ \Phi(a\_{k+1})\right] = & \text{"1"} / \ \text{"0"} \\ & \text{\$Fl|}\left[(k+1)\text{Gl}\mathbf{x}, \ p\text{Gl}\mathbf{x}, \ q\text{Gl}\mathbf{x}\right] = \ & \text{"1"} / \ \text{"0"} \\ & \text{\$x\\$B\!v}\_{k+1} = \ \text{"1"} / \ \text{"0"}, \ \text{Bew}(\mathbf{x}\_{k+1}) = \ \text{"1"} / \ \text{"0"}; \\ & \text{\$\Phi(a\_{p})||}\ \text{23}^{\text{\textquotedblleft}d\Phi\left(\overline{a\_{q,p,k+1}}\right)} \text{ & \text{\&}} \ \text{\$\Phi(a\_{p})||}\ \text{\$\tau^{\textquotedblright}\$} \text{(\$\overline{a\_{q,p,k+1}}\$)} = \ \text{"1"} / \ \text{"0"} \end{split}$$
