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Claudio Conforti

martes, 23 de octubre de 2012

A FUNDAMENTAL FLAW IN AN INCOMPLETENESS PROOF BY GEORGE BOOLOS , James R Meyer


This paper addresses a proof of incompleteness published by George Boolos. An
analysis of this proofs demonstrates that there is an elementary error in the proof;
the proof relies on the unproven assumption that the formal system can self-
reference its own formulas.

1 Introduction
Boolos published his proof of incompleteness in 1989 [2]. It is also to be found in two
books published subsequently [3, 4]. He claims that his proof is essentially di erent from
previously published proofs, since it operates on the principle of Berry's paradox.He claims
elsewhere [1] that his proof provides an essentially di erent reason for incompleteness.
Berry's paradox, of course, is a contradiction that is stated in natural language, which
means that it is stated with a lack of precision of de nition. One might expect that when
a corresponding statement is used as part of a mathematical proof, it would be stated
with su cient precision so that it is evident that there is not a hidden contradiction in
the statement.
However, Boolos's proof does not give a rigorously precise formulation of the Berry
paradox. Instead, he relies on a assumption that the formal system for which he claims
to prove incompleteness can self-reference its own formulas; Boolos provides no proof
of this assumption, and since his proof is entirely reliant on that assumption, his proof
cannot sensibly be called a proof at all. Rather than providing a di erent reason for
incompleteness, as Boolos claims, the proof merely demonstrates that once an assumption
is made that a formal system can have certain types of self-referential statements, the
result of incompleteness is a trivially obtained consequence.

2 Boolos's proof
The outline of Boolos's proof is as follows:
De nition: The language of arithmetic consists of 16 symbols: +, ×, 0, s, =, ¬, ,, -,
􀀀, , ¦, §, (, ), x, and œ.
Assumption: There is an algorithm M that outputs all true statements of the language
of arithmetic and no false ones.
De nition: [n] is the expression that consists of 0 preceded by n quantity of the symbols.
De nition: The formula F(x) is said to name the natural number n if the expression
¦x(F(x) x = [n]) is an output of the algorithm M.
Assumption: There is a formula in the language of arithmetic (i.e., using only the 16
symbols mentioned above) that states (by an appropriate encoding): `x
is a number that is named by some formula containing z symbols'. The
designation C(x;z) is used to refer to this formula.
De nition: B(x;y) is de ned as §z(z < y,C(x;z)).
De nition: A(x;y) is de ned as (¬B(x;y),¦a(a < x􀀀B(a;y))).
De nition: k is de ned as the `number of symbols in' A(x;y).
De nition: F(x) is de ned as §y(y = ([10]×[k]),A(x;y)).
Boolos then de nes a formula that is de ned in terms of this F(x) as ¦x(F(x) x = [n] ).
He asserts that this formula states that `x is the least number not named by any formula
containing fewer than 10k symbols' He then states that this formula cannot be an
output of the algorithm M, but that the formula is actually true. Boolos concludes
that this contradiction indicates that his initial assumption that there is an algorithm
M is incorrect, and that the contradiction proves that there is no such algorithm M.
That completes the proof.

3 Analysis of Boolos's proof
In this proof, Boolos ignores a basic tenet of logic that, in any proof by contradiction, the
contradiction indicates that at least one of the assumptions leading to that contradiction
is incorrect, but it does not specify which one.
Boolos's proof, besides the assumption that there is an algorithm M, assumes that there
is a formula C(x;z) of the language of arithmetic which encodes the expression `x is a
number that is named by some formula containing z symbols'. The formula that gives rise
to the contradiction, ¦x(F(x) x = [n] ), is de ned in terms of this formula C(x;z).
Boolos justi es his assumption regarding the formula C(x;z) by `sketching' the construction
of the formula, as follows:
\Let us now sketch the construction of a formula C(x;z) that says that x is a
number named by a formula containing z symbols. The main points are that
algorithms like M can be regarded as operating on `expressions', i.e., nite
sequences of symbols; that, in a manner reminiscent of ASCII codes, symbols
can be assigned code numbers (logicians often call these code numbers Godel
numbers); that certain tricks of number theory enable one to code expressions
as numbers and operations on expressions as operations on the numbers that
code them; and that these numerical operations can all be de ned in terms of
addition, multiplication, and the notions of logic."
Here Boolos correctly states that one can assign code numbers that correspond to the
16 symbols of the language of arithmetic. He correctly states that one can use the
coding of symbols to assign numbers that correspond to expressions of the language of
arithmetic. And that one can, for any operation on such expressions, by such encoding,
have a corresponding operation on the corresponding code numbers. And that any such
numerical operation can be de ned in terms of basic operations using the +, ×,¬, ,, -,
􀀀, , ¦, § operators. Boolos continues:
\Discussion of symbols, expressions (and nite sequences of expressions, etc.)
can therefore be coded in the language of arithmetic as discussion of the natural
numbers that code them. . . . tricks of number theory then allow all such talk
of symbols, sequences, and the operations of M to be coded into formulas of
arithmetic"
From the above, since each speci c expression of the language of arithmetic is encoded as
a speci c natural number, then it follows that in a statement of the meta-language that
refers to expressions of the language of arithmetic in general, there will be a variable in
that statement that has the domain of expressions of the language of arithmetic. It follows
that upon encoding that statement, the encoding will have a corresponding variable whose
domain is natural numbers.
For example, given the expression `s Proves t', where s and t are variables with the domain
of expressions of the language of arithmetic, then the encoding gives some relation of the
language of arithmetic R(x;y), where x and y are variables with the the domain of natural
numbers (in the format of the language of arithmetic), and which correspond to s and t
respectively.
It also follows that on decoding an expression of the language of arithmetic, since every
variable of the language of arithmetic has the domain of natural numbers, every free
variable of the language of arithmetic will decode to a free variable whose domain is
expressions of the language of arithmetic. For the example above, the relation R(x;y)
decodes to `s Proves t'.

So, Boolos's sketch outline regarding C(x;z) tells us that decoding the formula C(x;z)
gives an expression of the meta-language with two free variables, both of which have
the domain of expressions of the language of arithmetic. However, the expression which
Boolos asserts is the decoding of the formula C(x;z) is the expression `x is a number that
is named by some formula containing z symbols', which has two free variables x and z
which have the domain, not of expressions of the language of arithmetic, but of natural
numbers.
This demonstrates that Boolos's sketch of the construction of his formula C(x;z) fails to
substantiate his claim that that formula encodes `x is a number that is named by some
formula containing z symbols'. Boolos has failed to show that there are valid `tricks of
number theory' that can create an encoding of `x is a number that is named by some
formula containing z symbols' to an expression of the language of arithmetic, and which
also preserves the truth value of the expression.
Boolos's perfunctory justi cation only serves to introduce further assumptions, rather
than provide any logical clari cation. Boolos's claim that he has proved incompleteness
carries no logical validity whatsoever.
References
[1] George Boolos. A Letter from George Boolos. Notices of the American Mathematical
Society, 36:676, 1989.
[2] George Boolos. A New Proof of the Godel's Incompleteness Theorem. Notices of the
American Mathematical Society, 36:388{390, 1989.
[3] George Boolos. Logic, Logic, and Logic. Harvard University Press, 1999.
ISBN: 9780674537675.
[4] Reuben Hersh. What Is Mathematics, Really? Oxford University Press, 1997.
ISBN: 9780195113686.

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