Section 3 : Is Mendelson’s use of
the Gödel *β*-function
valid?

We derive a simple inconsistency from Mendelson’s proof of Gödel’s
*Theorem VI*. We trace the source of the inconsistency to an invalid
application of the Gödel *β*-function
for constructing a formula of the formal system that would *strongly*
represent a recursive function.

1 Gödel’s
*Theorem VI*

In his version[1]
of Gödel’s
*Theorem VI*[2],
Mendelson defines a primitive recursive relation W* _{1}*(

Since the relation W* _{1}*(

Mendelson then considers the well-formed formula :

(*i*) (*x** _{2}*) ~

Assuming *m* to be the Gödel-number
of the well-formed formula §1(*i*), he
substitutes the numeral ** m** for

(*ii*) (*x** _{2}*) ~

Since *m*
is the Gödel-number
of the well-formed formula §1(*i*), and §1(*ii*) comes from §1(*i*)
by substituting the numeral ** m** for the variable

(*iii*) W* _{1}*(

2 Mendelson’s version of Gödel’s proof

Mendelson then argues that :

(*1*)
Assume **S** is
consistent, and assume that ├** _{S}**
(

(*2*) Assume **S**
is ω-consistent[10],
and assume that ├** _{S}**
~ (

Mendelson thus concludes
that the sentence §1(*ii*) is undecidable in **S**.

3 Why is this reasoning ignored?

However, Mendelson also[13]
ignores the following straightforward reasoning that leads from his preceding
premises to a simple inconsistency in **S**.

We consider the well-formed formula :

(*i*) ~*W** _{1}*(

Assuming *r* to be the Gödel-number
of the well-formed formula §3(*i*), we
substitute the numeral ** r** for

(*ii*) ~*W** _{1}*(

Since *r*
is the Gödel-number
of the well-formed formula §3(*i*), and §3(*ii*) comes from §3(*i*)
by substituting the numeral ** r** for the variable

(*iii*) W* _{1}*(

We now note that :

(*1*)
Assume **S** is
consistent, and assume that ├** _{S}** ~

(*2*)
We thus have that ~*W** _{1}*(

We thus have the inconsistency in the meta-theory **M** of **S** :

The primitive
recursive formula ~*W** _{1}*(

The primitive recursive
formula ~*W** _{1}*(

5 Every primitive recursive formula is not expressible formally in a consistent S

Since **S**
is assumed consistent, the above establishes that the primitive recursive
formula W* _{1}*(

Mendelson’s assertion in his *Proposition 3.23*[18],
mirroring Gödel’s
implicit postulation in his *Theorem V* that *every* primitive
recursive formula is *strongly*[19]
representable in **S**,
is thus invalid.

6 Conclusion : The fallacy in Mendelson’s reasoning

The cause of the inconsistency is Mendelson’s invalid use of the
Gödel
*β*-function[20] for
constructing a formula of the formal system that would *strongly*
represent a recursive function.

Clearly Mendelson’s definition and development of the *β*-function
in his *Propositions 3.21-3.23* can be validly used, as explicitly proved
by him, to construct a formula representing a finite sequence of *undetermined*
length.

However, it cannot validly be used - as claimed by him without
proof[21]
- to construct a formula *strongly* representing a recursive function,
which is essentially an infinite sequence of *indeterminate* length, as
such *strong* representation involves implicitly defining an infinite
number of *β*-functions, a definition of doubtful validity[22].

Notes (*Why03bt.doc : 4/3/01 5:24:57 PM*)

[1]** **[Mendelson
1964 : *p142-144*]

[2]** **[Gödel
1931 : *p22*]

[3] Mendelson’s
formal system **S** differs significantly from that considered by
Gödel
in his seminal 1931 paper ‘*On Formally
Undecidable Propositions of Principia Mathematica and Related Systems I*’ [Gödel
1931 : *p9-13*], in that he includes
addition and multiplication as primitive undefined symbols. [Mendelson 1964 : *p102-103*]

[4]** **We
use bold lettering to denote the numeral ‘** u**’ in

[5] On the basis
of his Propositions 3.22 and 3.23 Mendelson postulates, without a rigorous
proof, that every primitive recursive function is strongly representable in **S**.

[6] We note that
Mendelson uses the notation ‘├_{S}** ***W** _{1}*(

[7]** **[Mendelson
1964 : *p143*]

[8] Proof of
proposition 3.31[Mendelson 1964 : *p143*]

[9] If we define
‘╟** _{A}**W

├_{S}** **(*x***_{2}**)

→** _{M}**
(

→** _{M}**
(

and :

├_{S}** **(*x***_{2}**)

**≡**** _{M}** ├

→** _{M}**
(

[10] ‘**S** is said to be ω-consistent if
and only if, for every well formed formula ** A**(

then it is not the case that ├** _{S}**
(

[11] See note 10 above.

[12] This meta-reasoning can be symbolically expressed as follows :

├_{S}** ****~ **(*x***_{2}**)

**≡**_{M}** **├_{S}** **(*E**x***_{2}**)

→_{M}**~**├_{S}** **(*x***_{2}**)

→** _{M}**
(

→** _{M}**
(

→** _{M}**
(

→_{M}**~**├_{S}** **(*E**x***_{2}**)

→_{M}**~**├_{S}** **(*E**x***_{2}**)

[13] The reference here is to ‘Is Gödel’s Platonist creation through definition valid?’ and ‘Why did Gödel ignore this reasoning?’.

[14] This meta-reasoning can be symbolically expressed as follows :

├_{S}** ****~ ***W** _{1}*(

→** _{M}**
(

→** _{M}**
(

and :

├_{S}** ****~ ***W** _{1}*(

→** _{M}**
(

[15] This can be
expressed symbolically in the meta-theory **M**
of **S** as **~**** **(*n*)├_{S}** ****~***W** _{1}*(

[16] We can express this implication by :

‘W* _{1}*(

[17] This meta-reasoning too becomes clearer if we express the preceding argument as :

**~**├_{S}** ****~***W** _{1}*(

→_{M}**~**(*n*)├_{S}** ****~***W** _{1}*(

→_{M}**~**(*n*)╟ _{All }_{A}**~**W* _{1}*(

→** _{M}**
(

→** _{M}**
├

[18]** **[Mendelson
1964 : *p134*]

[19]** **[Mendelson
1964 : *p118*]

[20] [Mendelson
1964 : *p131 Proposition 3.21*]

[21]** **[Mendelson
1964 : *p134*]

[22] Both Gödel’s
*Theorem V* and Mendelson’s *Proposition 3.23* implicitly assert, and
critically depend on, the postulated existence by definition of formal
expressions *in* **S**
(*i.e. functions expressible in terms of only the primitive symbols of ***S**) that are equivalent to
the primitive recursive functions *of* **S**
(*i.e. primitive recursive functions whose values are always expressible in
terms of only the primitive symbols of ***S**)
involved in the two hypotheses.

Such postulated existence by 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’.

**◄ Index
◄ Title ◄ Preface ◄ Contents ◄ Section 1 ◄ Section 2**

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

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