(This version finalised on Monday, August 6, 2001)
THE Gödel-number of every formula of "the system P" which contains a "number-sign" (or "numeral") must be greater than the numerical value of that number-sign itself.
For instance, if any formula of the system P contains the number-sign for the number 17, then its Gödel-number cannot possibly be equal to or less than 17.
This is because by definition, every such formula itself contains a number-sign, and this number-sign itself consists of a number of basic signs of the system P, this number of basic signs being one more than the natural number of which it is the number-sign.
Therefore the Gödel-number of that number-sign alone
must be greater than the numerical value of that
number-sign itself. (This does not even take into
consideration the rest of the formula!)
Let r be the Gödel-number of what Gödel calls a "class-sign" (i.e., a well-formed formula with a single free variable); and let that free variable be q. Then r is, essentially:
If we now define the function Sb (r q | Z(x)) as being the propositional formula obtained by the substitution, in the class-sign of which the Gödel-number is r, of the only free variable in it, q, by number-sign for the number x, then the sequence Sb (r q | Z(x)) becomes:
some-sequence-of-basic-signs_ƒƒƒ . . . s0_some-other-sequence-of-basic-signs
… where ƒ means "the successor of". Notice that there must be x+1 basic signs in the sequence ƒƒƒ... ƒ0.
However, by the system called Gödel-numbering, the basic sign ƒ is given a Gödel-number which is a non-zero natural number — and which is always, as a result, either 1 or greater than 1.
Thus the Gödel-number of the propositional formula Sb (r q | Z(x)) cannot possibly be equal to or less than x, since the number of basic signs in the sequence ƒƒƒ... ƒ0, as noted above, is x+1; and in addition there are the other basic signs in the formula, whose Gödel-numbers are also 1 or greater than 1, and which therefore contribute towards increasing the value of the Gödel-number of the formula Sb (r q | Z(x)).
In other words, the Gödel-number of any
propositional formula containing a number-sign — e.g.,
a formula resulting from the substitution of the
only free variable in a class-sign by a number-sign —
must always be greater than the numerical
value of that number-sign itself. Both its
Gödel-number and the numerical value of
the number-sign contained in it cannot possibly be
equal to one another.
The So-Called "Undecidable Formula"
Now let us assume, as our hypothesis, that there is in fact an undecidable propositional formula in the system P, and that g is its Gödel-number. By the above argument, the numerical value of g must be greater than the numerical value of the number-sign contained in it.
However, according to Gödel’s argument, the purported undecidable formula refers to itself; and thus g must also be the numerical value of the number-sign contained in the undecidable formula.
Therefore g cannot be the Gödel-number of any undecidable formula. Indeed it cannot be the Gödel-number of any propositional formula which refers to itself! If it were, it would contain a number-sign whose numerical value would be equal to g. And in that case the numerical value of that number-sign would have to be both exactly equal to and greater than g — which is logically impossible.
Since the implications of our hypothesis lead to an impossibility, the hypothesis itself cannot have been correct; and as a result, there can be no number g which is the Gödel-number of any "undecidable formula". Indeed there can be no number g which is the Gödel-number of any propositional formula that refers to itself.
(By the way: It may not be argued that the
so-called "undecidable formula" need not itself
contain its own Gödel-number, but may refer to
itself indirectly, by saying, in effect, "That
formula which is obtained by substituting the free
variable in formula number so-and-so by the number-sign
of its own Gödel-number is not provable". As
Gödel himself writes — in footnote No. 20 and
Definition No. 31 of his 1931 paper entitled "On
Formally Undecidable Propositions of Principia
Mathematica and Related Systems", wherein he tries
to prove his celebrated Theorem — such a formula,
containing as it must the sign "Subst" or "Sb" (standing
for the operation of substituting a variable in
a formula with a number) belongs to metamathematics,
not to mathematics. Thus if Gödel can
in fact prove that such an undecidable formula exists,
he can only prove thereby that mathematics by itself
cannot decide a metamathematical formula … to
which we should retort, as any school-boy justifiably
might: "Big deal!")
If you have any
comments, please e-mail
I got the idea for this argument after reading an article entitled On the Gödel’s Formula by Jailton C. Ferreira, which can also be downloaded in .pdf (Adobe Acrobat) format from Cornell University's following web page:
Of course Gödel-numbers themselves belong
to metamathematics, and not to mathematics, and may not
validly be used in any mathematical formula. Any formula
in which a Gödel-number appears must belong to
metamathematics, not to mathematics; and thus if
Gödel can actually prove that there is
such a formula and that it is indeed
undecidable, all he can possibly prove thereby is that
it is his metamathematics that is incomplete ...
leaving mathematics itself as complete as ever
One can hardly argue that mathematics and
metamathematics are essentially the same thing: for if
they were, it should be possible to derive all
of metamathematics from the axioms of mathematics alone
(such as the Peano axioms, or the axioms of Zermelo and
Fraenkel, later extended by John von Neumann). But no
one, not even Gödel, has ever been able to lay
claim to having performed such a superhuman feat.
There is a much fuller, but still
comprehensible, account of the above ideas in my
of Gödel's 1931 Paper Entitled "On Formally
Undecidable Propositions of Principia
Mathematica and Related Systems",which I
wrote last year in collaboration with Ferdinand
Romero, and which is available for download in .pdf
format from my Home Page.