Logic and Language

Logic and Language

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

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Various people have tried to get round my demonstration of the flaw in Gödel’s Proposition V of Gödel’s original incompleteness proof by claiming that the R(*x*_{1} … *x*_{n}) in the proposition is not actually a number-theoretic relation, but is only an expression that * refers* to an independently “existing” number-theoretic relation that does not belong to any language. This of course is a Platonist notion that has no evidence nor logic to support it. Nevertheless, it is easily shown that this notion makes no difference whatsoever to the demonstration that Gödel’s proof is irredeemably flawed, as shown below.

So, following that Platonist notion, we will suppose that an expression that includes numbers and variables for numbers cannot itself be a number-theoretic relation, but simply an expression that corresponds to a number-theoretic relation that “exists” independently of any language. That being the case, then it must also follow that there are valid expressions of the formal system that correspond to such independently “existing” number-theoretic relations.

That being so, let’s look at Gödel’s Proposition V in Gödel’s original incompleteness proof. Clearly, for any expression R(*x*_{1} … *x*_{n}), there is also a corresponding formula *f*(*x*_{1} … *x*_{n}) of the formal system that corresponds to the same independently “existing” number-theoretic relation that R(*x*_{1} … *x*_{n}) corresponds to. That being the case, then there is no reason why we should not use this formula *f*(*x*_{1} … *x*_{n}) rather than R(*x*_{1} … *x*_{n}) to refer to that independently “existing” number-theoretic relation. This gives us, for Gödel’s Proposition V, for a number-theoretic relation with one free variable:

For every recursive relation *f*(*x*_{1}) … :

*f*(*x*_{1}) ⇒ Bew{Sb[r ^{u1}/_{Z(x1)} ]} …

This is a proposition, and *x*_{1} is a variable, whereas *u*_{1} is not a variable.
The uninitiated may think that *u*_{1} is a variable because Gödel refers to it as a *free variable*, but what one must recognize is that when Gödel uses word *free variable* in italics (in Meltzer’s translation; in Van Heijenoort’s and Hirzel’s translations capital letters are used for the same purpose), that actually means the specific number that is defined to correspond to that free variable - so in this case, the *free variable* *u*_{1} is not any number, but a specific prime number greater than 13. Gödel states that this number can be arbitrarily chosen - in other words, we can choose any variable of the formal system, and there will be a prime number greater than 13 that corresponds to it. That number is given by a function that Gödel describes in his paper, and we will refer to this function by U(X), (Footnote: Gödel did not give this function a name, but his description clearly describes a function.) where X is a variable whose domain is variables of the formal system, i.e., *x*_{1} … *x*_{n} and where U(X) evaluates as a prime number greater than 13. We will choose *x*_{1} to be the variable of the formal system. This now gives us for Proposition V:

For every recursive relation *f*(*x*_{1}) … :

*f*(*x*_{1}) ⇒ Bew{Sb[r ^{U(x1)}/_{Z(x1)} ]} …

But now we have a major problem. In the expression:

*f*(*x*_{1}) ⇒ Bew{Sb[r ^{U(x1)}/_{Z(x1)} ]} …

the *x*_{1} is a free variable, with the domain of all natural numbers. But the function U is only defined for values that are variables of the formal system, and it is not defined if an attempt is made to substitute a numerical value for the *x*_{1} in U(*x*_{1}).

Note that an attempt to claim that you can substitute the expression U(*x*_{1}) by the number 17 does not obviate the problem, since in mathematics, it is fundamental that if U(*x*_{1}) = 17, then that equality always exists, and the two expressions must be always interchangeable. And if they are interchangeable, then *x*_{1} is a free variable wherever it occurs in the expression:

*f*(*x*_{1}) ⇒ Bew{Sb[r ^{U(x1)}/_{Z(x1)} ]} …

or, if you insist on Platonist notions, the *x*_{1} corresponds to an independently “existing” free variable in the independently “existing” relation that corresponds to the above expression. But it should be abundantly clear by now that such Platonist notions only add verbiage that has no substance.

Yet again, no matter which way you look at Gödel’s Proposition V, if you analyze it logically, you find that there is always a logical absurdity within it.

See also the page Gödel’s Proposition V.

Footnotes:

Diverse opinions and criticisms are welcome, but messages that are frivolous, irrelevant or devoid of logical basis will be blocked. Difficulties in understanding the site content are usually best addressed by contacting me by e-mail. Note: you will be asked to provide an e-mail address - any address will do, it does not require verification. Your e-mail will only be used to notify you of replies to your comments - it will never be used for any other purpose and will not be displayed. If you cannot see any comments below, see Why isn’t the comment box loading?.

There is now a new page on a contradiction in Lebesgue measure theory.

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.

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

https://www.jamesrmeyer.com