Section 1 : Is Gödel’s Platonist creation through definition valid?

Gödel argues that every recursive
function can be expressed in his formal system of Arithmetic S in terms of the primitive
symbols of S. We argue,
however, that the recursive function ‘*x**+**y*’ cannot be expressed formally in terms
of the primitive symbols of S.

1 Gödel’s
*Theorem V*

In *Theorem V*[1]
of his seminal 1931 paper ‘*On Formally Undecidable Propositions of Principia
Mathematica and Related Systems I*’, Gödel
proposes the hypothesis - and supports it with an argument by induction - that
:

*a*) every *n*-ary arithmetical recursive[2] relation is constructively[3] expressible as a formula[4] of his formal system[5] **S** in an intuitionistically unobjectionable way[6];

*b*) every *n*-ary arithmetical recursive relation can be
constructively associated with a unique Gödel-number[7] in an intuitionistically unobjectionable
way.[8]

2
Gödel’s reasoning in *Theorem V*

(*a*)
In his proof of the above theorem, Gödel
considers ‘… all relations ** R**(

(*b*)
Gödel’s argument is that if we apply complete induction on the rank of *φ*, then
‘… the theorem is trivial for functions of rank one (*i*.*e*. *constants*
*and the* *function* ‘*x*
+ 1’)’.[10]

(*c*)
Further, assuming *φ* to have the rank *m*,[11]
Gödel
notes that ‘… *φ* results from functions of lower
rank *φ** _{1}*,

(*d*)
Gödel
then argues that ‘… since, by inductive hypothesis, everything is already
proved for *φ** _{1}*,

(*e*)
He notes that ‘… the definitional procedures by which *φ* arises[15]
from *φ** _{1}*,

(*f*)
He concludes that one must therefore obtain ‘… from *r** _{1}*,

3 Applying Gödel’s
reasoning to ‘*x*+*y*’

(*a*)
We now apply Gödel’s reasoning to the recursive relation ** R**(

(*b*)
We note that the definition of *sum*(*x*, *y*)
can be formally imitated[18]
in the formal system **S**
by ‘**sum**(*x*, 0) ≡ *x*; **sum**(*x*, *N**y*) ≡ *N***sum**(*x*, *y*)’, where ‘*N**x*’
is the successor function[19]
of rank one in **S**.

(*c*)
Now Gödel’s
thesis is that a formal expression consisting of only the primitive symbols of **S** and corresponding to the
formal recursive function ‘**sum**(*x*, *y*)’, of rank two, can be
constructively built up from the formal expression ‘*N**x*’, of rank one, by the formal operations
of substitution or recursive definition.[20]

(*d*)
Gödel’s
conclusion then is that, from the Gödel-number
*r** _{1}*=22.316 of ‘

4 ‘sum(*x*, *y*)’
cannot be directly expressed in S

We define *N *^{[y]}0 ≡ *y* so that, for example, *N** *^{[}^{3}^{]}0 ≡ ** NNN**0 and ‘

**sum**(*x*, *y*) ≡ **sum**(*x*,* **N**
*^{[y]}0) ≡ *N***sum**(*x*,* **N**
*^{[y}^{-}^{1}^{]}0) ≡ *N**
*^{[y]}*x*

For any given *k*, **sum**(*x*, *k*) is clearly expressible formally in terms of only the primitive symbols
‘** N**’ and ‘

However, for a variable *y*,
we cannot obviously express **sum**(*x*, *y*) - or the equivalent *N*^{[y]}*x* - in
terms of only the primitive symbols of **S**[23].
Similarly, for a variable *y*, *r _{y }*too
cannot be obviously enumerated so as to yield a unique Gödel-number

5 Platonist creation through definition

Hence a formal expression consisting of only the primitive
symbols of **S** and
corresponding to the recursive function ‘*sum*(*x*, *y*)’, i.e. ‘*x*+*y*’, cannot be constructively
built up[24]
from the recursive formal function ‘*N**x*’ by ‘*imitating*’ the formal operations of
substitution or recursive definition within **S**.[25]

Thus Gödel’s thesis in §3(*c*) and his conclusion[26]
in §3(*d*) assert, and critically depend on,
the *indirect* existence through definition of a formal expression in
terms of only the primitive symbols of **S**
that is formally equivalent to the formal recursive function ‘**sum**(*x*, *y*)’ defined in §3(*b*) above.

Such existence through definition is strongly reminiscent of the
Platonist postulation and creation by definition of a paradoxical set ‘*Russell*’
within a Set Theory whose elements are all, and only, those sets of the Theory
that do not belong to themselves; or the similar postulation and creation by
definition of a paradoxical ‘*Liar*’ sentence within the English language
which reads as ‘The ‘*Liar*’ sentence is a lie’.

Despite the absence of a rigorous proof in his paper in support of his postulation by definition Gödel, reportedly a ‘…very strong Platonist.’[27], was apparently comfortable that this lacuna in his reasoning[28] would not yield an inconsistency.

Although formal number theory, such as developed in Mendelson[29],
attempts a justification for Gödel’s
reasoning by using the stratagem of a ‘Gödel
*β*-function’, this only permits sequences of *determinate*, if *undetermined*,
length to be represented by formulas of a formal system **S '** where addition is
an undefined primitive symbol.

Despite
Mendelson’s claim - also without a rigorous proof[30]
- the Gödel
*β*-function cannot, however, be used to validly construct a formula *strongly*[31]
representing a primitive recursive function, which is essentially an infinite
sequence of *indeterminate* length.

Moreover, in companion papers[32],
we show how inconsistencies can arise in both Gödel’s
and Mendelson’s formal systems out of the postulation that every primitive
recursive function can be *strongly* represented in their systems.

Notes (*Why01et.doc : 4/3/01 5:23:36 PM*)

[1] ‘For every
recursive relation* *** R**(

** R**(

~* *** R**(

We note the implicit implication here that *r* is a specific
natural number - the Gödel-number of an
implicitly postulated formal expression corresponding to the recursive formula ** R**(

[2] ‘A
number-theoretic function *φ* is said to be recursive if there exists a finite sequence of
number-theoretic functions *φ _{1}*,

[3] i.e. by a
finite procedure. [Gödel 1931 : *p26 footnote
45a*].

[4] Gödel
defines the primitive symbols of **S**
as consisting of various constants and variables; combinations of certain
symbols as terms; and combinations of certain symbols and terms as the various
types of formulas of **S**.
[Gödel
1931 : *p11*]

[5] Gödel
develops the formal system **S**
as ‘ … essentially the system which one obtains by building the logic of PM (*Principia
Mathematica*) around Peano’s axioms (numbers as individuals, successor
relation as undefined primitive concept).’ [Gödel
1931 : *p10*]

[6] i.e. without
use of existence proofs such as arguments that depend, for example, on the use
of the Law of the Excluded Middle. [Gödel
1931 : *p26 footnote 45a*]

[7] Gödel
describes in detail how every primitive symbol of **S**, and every formal expression consisting of a
finite sequence of such symbols of **S**,
can be arithmetised and correlated uniquely to some natural number - which we
define as the Gödel-number of the formal
expression in question. [Gödel 1931 : *p13*]

[8] §1(*a*) and §1(*b*) are,
essentially, Gödel’s *Theorem V*,
which he notes as the rigorous expression of the ‘ … fact which can be vaguely
formulated as the assertion that every recursive relation is definable within
the system **S** (under
its intuitive interpretation) … without reference to the intuitive meaning of
the formulas of **S**.’
[Gödel
1931 : *p22*]

[9] [Gödel
1931 : *p22-23*]

[10] Using the Gödel-numbering
system as defined by Gödel in Davis 1965 : *p53*,
we determine in the first case the formal expression in Gödel’s
formal system **S** corresponding
to say the recursive relation ‘*x* = 2’ is ‘*x* = ** NN**0’, and its Gödel-number
is

[11] We note
that the premise in Gödel’s mathematical
induction hypothesis is that ‘*m*’ and ‘*k*’ are not variables but *determinate*, even if *undetermined*,
natural numbers.

[12] [Gödel
1931 : *p15*]

[13] i.e. Gödel-numbers.

[14] i.e. the
recursive functions *φ _{1}*,

[15] Gödel’s
implicit belief in the constructibility of such a formula *φ* within **S** by such definition is reflected in his
initial overview of his paper, where he notes that ‘… as soon as one has
obtained the formula *φ*, one can, of course, also
determine the number *r*,
and thereby effectively write down the undecidable proposition itself.’ [Gödel
1931 : *p8 footnote 13*]

[16] We
re-emphasise here that this ‘imitation’ holds only for *determinate*, if *undetermined*,
‘*m*’ and
‘*k*’.
Further, that the aim of Gödel’s reasoning in his
Theorem *V* is to eventually establish that the ‘imitation’ in the formal
system **S** of the
above procedure also holds for an *indeterminate* - i.e. free variable - ‘*x*’.

[17] We note
that Gödel
here assumes the constructibility of such a Gödel-number
*r* without
proof. Thus he remarks that ‘… when this proof is rigorously carried out, *r* will naturally
not be defined by this shortcut through the intuitive interpretation, but
rather by its purely formal structure.’ [Gödel
1931 : *p22 footnotes 38 and 41*]

[18] [Gödel
1931 : *p23*]

[19] [Davis 1965
: *p46*]

[20] [Gödel
1931 : *p15*]

[21] We use the
Gödel-numbering
system as defined by Gödel in Davis 1965 : *p53*.

[22] [Davis 1965
: *p53*]

[23] What we
have shown here is that although §3(*b*) does define a formal recursive
function ‘sum(*x*, *y*)’ *of* **S**, this function cannot be
expressed as a formal expression *in* **S**
in terms of only the primitive symbols of **S**.

[24] Dr. Chetan
Mehta first pointed this out to the author in private correspondence. We also
show elsewhere that every recursive relation is not expressible formally in a
simply consistent formal system. [Anand 2000 : *chapters 2*,* 3 & 5*]

[25] Thus,
though the function ‘*N**x*’ is a formal expression of **S** and easily Gödel-numbered,
a recursive definition of the *form* of the function ‘sum(*x*, *N**y*)’ is essentially self-referential. Clearly, the *form*
of ‘sum(*x*, *N**y*)’ - as represented by some formal expression of **S** and distinct from its
recursively defined formal value - is Gödel-numberable
if and only if the *form* of the function ‘sum(*x*, *y*)’ - as represented by some formal expression of **S** - is Gödel-numberable.

[26] That Gödel
had reservations about the validity of his argument and conclusion is reflected
in his subsequent lectures, where he notes that ‘ … this proof is not necessary
for the demonstration of the existence of undecidable arithmetic propositions
in the system considered. For, if some recursive function were not
‘represented’ by the corresponding formula constructed on pages 58-59, this
would trivially imply the existence of undecidable propositions unless some
wrong proposition on integers were demonstrable’. [Davis 1965 : *p59 footnote
21*]

[27] [Penrose 1990 : *p147*]

[28] Perhaps since he depended on Whitehead and
Russell’s ‘… theory of types as our means for avoiding paradox.’. [Davis 1965 :
*p46*]

[29] [Mendelson 1964 : *chapter 3*,* p102*]

[30] [Mendelson 1964 : *p134*]

[31] [Mendelson 1964 : *p118*]

[32] ‘Why did Gödel ignore this
reasoning?’ and ‘Is Mendelson’s use of the Gödel
*β*-function
valid?’.

**◄ Index
◄ Title ◄ Preface ◄ Contents ▲ **Section 1

*Section
3* ► *Section 4*
► *References* ► *Roots* ► *Bibliography* ► *Web_links* ►

** ( Updated : 10/11/01 1:57:50 AM by re@alixcomsi.com)**