Copyright © James R Meyer 2012 - 2017 www.jamesrmeyer.com
This is the point in Gödel’s proof where we need a new function that is similar to the Gödel numbering function. The Gödel numbering function is not a number relationship, since it refers to symbols of the formal language that aren’t numbers, such as the symbol for equals, the symbol for plus, and so on.
We might suppose that we say that we’re going to use the Gödel numbering function only for symbols of the formal language that actually are numbers. So, if we call the Gödel numbering function GN(x), we say that we can only substitute the x by symbols of the formal language that represent numbers.
However, that still doesn’t make GN(x) into a number relationship – because the actual definition of the Gödel numbering function still refers to symbols other than numbers. In order for his proof to work, Gödel needs a function that is similar to the GN(x) function, but which is a purely number relationship.
The idea is that this new function will only take number values for its free variable, and provided you’re only inputting numbers, and the function only results in number values, this new function can do exactly the same thing as the Gödel numbering function. Since this new function doesn’t refer to anything other than numbers, and variables for numbers, and is defined only in terms of numbers, then it is a number relationship.
We can call this new function anything we like, but I’m going to call it the Basic Numbering function, or BN for short, or BN(x), where x is its free variable. (Footnote: In Gödel’s original proof, Gödel calls this function Z(n) - it is his ‘relation’ 17.) That means that whenever we put in symbols that are the symbols for a number, this BN function gives us the correct Gödel number – so we can get a Gödel number for any number we want.
You might wonder what happens if we try to substitute the variable of BN(x) by a symbol that isn’t a number. The answer is simple – the definition of BN(x) does not include any reference to specific values that are not number values. So trying to substitute a non-number value wouldn’t make any sense. It would be like trying to use any foreign word in a language and expecting it to make sense. With this function BN(x), wherever it appears, it has to be the case that only numbers can be substituted for its free variable x.
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It has come to my notice that, when asked about the demonstration of the flaw in his proof (see A Fundamental Flaw in an Incompleteness Proof by Peter Smith PDF), Smith refuses to engage in any logical discussion, and instead attempts to deflect attention away from any such discussion. If any other reader has tried to engage with Smith regarding my demonstration of the flaw, I would be interested to know what the outcome was.
There is a new addition to the page Yet another flawed incompleteness proof, where Berto’s proof of incompleteness in his book There’s something about Gödel comes under scrutiny.
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.
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 - 2017