A Simplified Explanation of Gödel’s proof - Part 7
Part 7: Another Numbering System
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.
As site owner I reserve the right to keep my comments sections as I deem appropriate. I do not use that right to unfairly censor valid criticism. My reasons for deleting or editing comments do not include deleting a comment because it disagrees with what is on my website. Reasons for exclusion include:
Frivolous, irrelevant comments.
Comments devoid of logical basis.
Comments with excessive number of different points.
Questions about matters that do not relate to the page they post on. Such posts are not comments.
Comments with a substantial amount of mathematical terms not properly formatted will not be published unless a file (such as doc, tex, pdf) is simultaneously emailed to me, and where the mathematical terms are correctly formatted.
Reasons for deleting comments of certain users:
Bulk posting of comments in a short space of time, often on several different pages, and which are not simply part of an ongoing discussion. Multiple anonymous usernames for one person.
Users, who, when shown their point is wrong, immediately claim that they just wrote it wrong and rewrite it again - still erroneously, or else attack something else on my site - erroneously. After the first few instances, further posts are deleted.
Users who make persistent erroneous attacks in a scatter-gun attempt to try to find some error in what I write on this site. After the first few instances, further posts are deleted.
Difficulties in understanding the site content are usually best addressed by contacting me by e-mail.
Note: a password enables editing of comments, an email enables notification of replies