Gödel's Theorem by (This version finalised on Monday, August 6, 2001)
Argument THE Gödelnumber of every formula of "the system P" which contains a "numbersign" (or "numeral") must be greater than the numerical value of that numbersign itself. For instance, if any formula of the system P contains the numbersign for the number 17, then its Gödelnumber cannot possibly be equal to or less than 17. This is because by definition, every such formula itself contains a numbersign, and this numbersign 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 numbersign. Therefore the Gödelnumber of that numbersign alone
must be greater than the numerical value of that
numbersign itself. (This does not even take into
consideration the rest of the formula!) Example Let r be the Gödelnumber of what Gödel calls a "classsign" (i.e., a wellformed formula with a single free variable); and let that free variable be q. Then r is, essentially: somesequenceofbasicsigns_q_someothersequenceofbasicsigns If we now define the function Sb (r q  Z(x)) as being the propositional formula obtained by the substitution, in the classsign of which the Gödelnumber is r, of the only free variable in it, q, by numbersign for the number x, then the sequence Sb (r q  Z(x)) becomes: somesequenceofbasicsigns_ƒƒƒ . . . s0_someothersequenceofbasicsigns … 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 Gödelnumbering, the basic sign ƒ is given a Gödelnumber which is a nonzero natural number — and which is always, as a result, either 1 or greater than 1. Thus the Gödelnumber 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 Gödelnumbers are also 1 or greater than 1, and which therefore contribute towards increasing the value of the Gödelnumber of the formula Sb (r q  Z(x)). In other words, the Gödelnumber of any
propositional formula containing a numbersign — e.g.,
a formula resulting from the substitution of the
only free variable in a classsign by a numbersign —
must always be greater than the numerical
value of that numbersign itself. Both its
Gödelnumber and the numerical value of
the numbersign contained in it cannot possibly be
equal to one another. The SoCalled "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 Gödelnumber. By the above argument, the numerical value of g must be greater than the numerical value of the numbersign 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 numbersign contained in the undecidable formula. Therefore g cannot be the Gödelnumber of any undecidable formula. Indeed it cannot be the Gödelnumber of any propositional formula which refers to itself! If it were, it would contain a numbersign whose numerical value would be equal to g. And in that case the numerical value of that numbersign 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 Gödelnumber of any "undecidable formula". Indeed there can be no number g which is the Gödelnumber of any propositional formula that refers to itself. (By the way: It may not be argued that the
socalled "undecidable formula" need not itself
contain its own Gödelnumber, but may refer to
itself indirectly, by saying, in effect, "That
formula which is obtained by substituting the free
variable in formula number soandso by the numbersign
of its own Gödelnumber 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 schoolboy justifiably
might: "Big deal!") Comments If you have any
comments, please email
me. Acknowledgement 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:
Postscript Of course Gödelnumbers themselves belong
to metamathematics, and not to mathematics, and may not
validly be used in any mathematical formula. Any formula
in which a Gödelnumber 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
it was! PostPostscript 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. PostPostPostscript There is a much fuller, but still
comprehensible, account of the above ideas in my
book Critique
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.
