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 (note: 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.
Diverse opinions and criticisms are welcome, but messages that are frivolous, irrelevant or devoid of logical basis will be blocked (comments will be checked before appearing on this site). 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 - this will only be used to notify you of replies to your comments - it will never be used for any other purpose, will never be displayed and does not require verification. Comments are common to the entire website, so please indicate what section of the site you are commenting on.
If you cannot see any comments below, it may be that a plug-in on your browser is blocking Disqus comments from loading. Avast anti-virus in particular is known to do this, especially with Internet Explorer and Safari. See Disqus Browser plug-in/extension conflicts or Why isn’t the comment box loading?.
Please wait for comments to load …
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:
All pages on this website are printer friendly, and will print the main content in a convenient format. Note that the margins are set by your browser print settings.
Comments on this site are welcome, please see the comment section.
Please note that this web site, like any other is a collection of various statements. Not all of this web site is intended to be factual. Some of it is personal opinion or interpretation.
If you prefer to ask me directly about the material on this site, please send me an e-mail with your query, and I will attempt to reply promptly.
Feedback about site design would also be appreciated so that I can improve the site.
Copyright © James R Meyer 2012 - 2017