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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 < xB(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.

Etiquetas:
proof of incompleteness - Boolos

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