Logic and Language

Logic and Language

Copyright © James R Meyer 2012 - 2018 www.jamesrmeyer.com

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In Gödel’s proof of incompleteness (see online English translation of Gödel’s original proof or English translation of Gödel’s original proof, PDF file) there is an arbitrary distinction between number-theoretic relations and formulas of the formal system.

In the proof, a symbol sequence such as “*x* + *SSSSSS*0 > *SSS*0” is simply a sequence of symbols that belongs to the formal system, and “*x* + *SSSSSS*0 > *SSS*0” is neither a number-theoretic relation, nor can it be a name that refers to an abstract number-theoretic relation.

But on the other hand an expression such as “*x* + 6 > 3” is either a number-theoretic relation, or else a name that refers to the abstract concept that is an abstract number-theoretic relation.

However, Proposition V in Gödel’s proof refers in a vague way to some sort of isomorphism between abstract number-theoretic relations and formulas of the formal system, by simply asserting that there is a “correspondence” between every number-theoretic relation that is recursive and some formula of the formal system. Of course there is no essential difference between “*x* + 6 > 3” and “*x* + *SSSSSS*0 > *SSS*0”; they simply belong to slightly different systems. And that simply means that, given a language that is a meta-language to both systems, there can be a translation function in that meta-language that translates the symbol sequences of of system to symbol sequences of the other system.

And that means that if “*x* + 6 > 3” is simply a name that refers to the abstract concept that is an abstract number-theoretic relation, then “*x* + *SSSSSS*0 > *SSS*0” is also a name that refers to the very same abstract concept.

Or if the symbol sequence “*x* + 6 > 3” satisfies the definition of a number-theoretic relation, then the symbol sequence “*x* + *SSSSSS*0 > *SSS*0” also satisfies that definition.

Gödel’s paper doesn’t define “number-theoretic relation”; and Gödel’s perfunctory outline of how a proof of his Proposition V might be created relies on the pretense that formulas of the formal system are not number-theoretic relations, nor are they names for the same abstract concept as number-theoretic relations. Without that pretense, his proof outline results in a blatant contradiction as shown below (see also the paper The Fundamental Flaw in Gödel’s Proof of the Incompleteness Theorem PDF ):

Gödel’s Proposition V is expressed in terms of variables *x*_{1}, *x*_{2}, …, *u*_{1}, *u*_{2}, … . But, as for any other mathematical proposition, one can use different variable names in a proposition, provided they do not clash with variable names that are already in that proposition.

In Gödel’s proof we have the Gödel numbering function defined as Φ(*s*_{2}), where *s*_{2} is a symbol sequence of the formal system. He also defines a variable numbering function which assigns a number to every variable of the formal system. Gödel does not give this function a name, but for convenience we shall assign the name *β*(*s*_{1}) to it, where *s*_{1} is a variable of the formal system.

Gödel’s Proposition V is (here stated in terms of relations with only one free variable):

R(*x*_{1}) ⇒ Bew{Sb[*r*, *u*_{1}, Z(*x*_{1})]}

¬R(*x*_{1}) ⇒ Bew{Neg Sb[*r*, *u*_{1}, Z(*x*_{1})]}

The terms *relation-sign* and *free variable* have special meanings in Gödel’s proof. The term *relation-sign* *r* means that *r* is a number where *r* = Φ(*s*_{2}), where *s*_{2} is a formal system formula. The term *free variable* *u*_{1} means that *u*_{1} is a number where *u*_{1} = *β*(*s*_{1}) , where *s*_{1} is a formal system variable. So, including these definitions explicitly in the statement of the proposition does not change the assertion of the proposition in any way:

R(*x*_{1}) ⇒ Bew{Sb[*r*, *u*_{1}, Z(*x*_{1})]}

¬R(*x*_{1}) ⇒ Bew{Neg Sb[*r*, *u*_{1}, Z(*x*_{1})]}

Now, since there is no variable *v*_{1} already in the proposition, we may use *v*_{1} instead of *x*_{1} as the variable name in the above, giving:

R(*v*_{1}) ⇒ Bew{Sb[*r*, *u*_{1}, Z(*v*_{1})]}

¬R(*v*_{1}) ⇒ Bew{Neg Sb[*r*, *u*_{1}, Z(*v*_{1})]}

Now, take any R(*v*_{1}) as referred to by the proposition, and translate it into a formula of the formal system. Either that formula is a recursive relation, or else it is a name that refers to the abstract concept that is the same abstract number-theoretic relation as the R(*v*_{1}) referred to originally.

So there will be such formulas of the formal system for which Proposition V holds. Let’s call one such formula R_{f} (*v*_{1}), where *v*_{1} is the free variable of that formula. So for that formula, we have:

R_{f} (*v*_{1}) ⇒ Bew{Sb[*r*, *u*_{1}, Z(*v*_{1})]}

¬R_{f} (*v*_{1}) ⇒ Bew{Neg Sb[*r*, *u*_{1}, Z(*v*_{1})]}

Now, according to Gödel, the number *u*_{1 }(where *u*_{1} = *β*(*s*_{1})) may be chosen arbitrarily (provided that *s*_{1} is a formal system variable). So, let’s choose the number 17, which corresponds by Gödel variable numbering to the variable *v*_{1} of the formal system. This gives us:

R_{f} (*v*_{1}) ⇒ Bew{Sb[*r*, *u*_{1}, Z(*v*_{1})]}

¬R_{f} (*v*_{1}) ⇒ Bew{Neg Sb[*r*, *u*_{1}, Z(*v*_{1})]}

This is, of course, ridiculous, since in the above *“ proposition”*, we have that

This show the nonsensical nature of Gödel’s Proposition V, and shows why it was absurd for Gödel to simply assume that it was a valid proposition, and to assume that it could be easily proved. It isn’t a valid proposition, it’s complete nonsense where there is a confusion of language and meta-language.

The language confusion arises because the proposition implies a function that translates from “number-theoretic relations” to formal system formulas (the correspondence referred to in the proposition), but if such a function is used, then that function is a meta-language to both “number-theoretic relations” and formal system formulas. And that means that in that function all symbols and symbol sequences of both “number-theoretic relations” and formal system formulas are objects as seen by that function, and therefore they must be objects in the proposition, and hence they cannot be variables in the proposition.

The only way to attempt to get around this absurdity is to fudge the issue, and to pretend that there is some distinction of language level between number-theoretic relations and formal system formulas, even though a translation function prevents that being possible.

See also the page The Platonist response to the flaw in Gödel’s proof. The paper The Fundamental Flaw in Gödel’s Proof of the Incompleteness Theorem PDF examines Gödel’s Proposition V in more detail.

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There is now a new page Halbach and Zhang’s *Yablo without Gödel* which analyzes the illogical assumptions used by Halbach and Zhang.

I found that making, adding or deleting footnotes in the traditional manner proved to be a major pain. So I developed a different system for footnotes which makes inserting or changing footnotes a doddle. You can check it out at Easy Footnotes for Web Pages (Accessibility friendly).

I have now added a new section to my paper on Russell O’Connor’s claim of a computer verified incompleteness proof. This shows that the flaw in the proof arises from a reliance on definitions that include unacceptable assumptions - assumptions that are not actually checked by the computer code. See also the new page Representability.

There is now a new page on Chaitin’s Constant (Chaitin’s Omega), which demonstrates that Chaitin has failed to prove that it is actually algorithmically irreducible.

8 Apr 2016 Are we alone in the Universe?

13 May 2015 Good Math, Bad Math?

31 Mar 2015 Cranks and Crackpots

16th Mar 2015 Bishops Dancing with Pixies?

For convenience, there are now two pages on this site with links to various material relating to Gödel and the Incompleteness Theorem

– a page with general links:

– and a page relating specifically to the Gödel mind-machine debate:

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Copyright © James R Meyer 2012 - 2018

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