Godel numbering is the SRH.

Update: 6/6/2026

This is importantly wrong.

Godel Numbering is expressible within Principia!

This means that within Principia it is possible to construct diagonalisation so that when Godel's famous statement says it is not provable this is not a piece of text that the reader understand to be self referential.

Rather it is a piece of text that lays out logical steps that link the statement to itself, and in particular logical steps that link the statement to a statement it is not provable. So every part of the Theorem is derived by the axioms and rules of the logic. The reader or computer of Godel's Theorem is just following the rules.

Now it is true that how the reader or the computer work is outside the system, but as long as they can encode the rules it is enough to express the proof and the devastating conclusion that Consistency is impossible because the system can prove a contradiction, and if we accept that it is true then Incompleteness as the statement says it is not provable.

So my original post was a meta level short. Godel Numbering was indeed encoded within Principia and PM can indeed mechanically generate every step of Godel's proof without insight from the computer.

To illustrate: the set {3,6,1,4} does indeed have a 4 at the 4th index but this coincidence depends upon the reader/computer to notice. It is not something in the system.

Godel by contract modelled this meta level so that the system itself could express the link. Thus diagonalisation was not a meta feature as Cantor had argued for his infinite ascent, but was now within the very system under discussion. So Godel's work is genuinely a meta level above, and his diagonalisation is both a feature of the system, and encoded within the system.

 

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Godel famous statement G to recap says:

Given formula:

F: ~(∃x)(∃y)(Pxy · Qzy) with Godel number f and free variable z

If we apply the formula to itself by binding f to z then we get the famous Godel statement

G: ~(∃x)(∃y)(Pxy · Qfy) see

If we say that the godel number provides a name for the sentences so sentence X has a godel number x whose physical numerals can be used as a name like "x", So it states that there is no string "x" which proves the statement "y" when y is the godel number of this formula (that is what the Qfy is a way of determining) i.e. "y" is a shorter name of "~(∃x)(∃y)(Pxy · Qfy)". If this were ever derived within PM, the system would have found a string "x" which proved "y". This proof of a contradiction would invalidate PM and allow for any string to be proved (this being the danger of contradiction). On the other hand if it can't be proved then it is true. But this extraordinary result shows there are things which are true which lie outside the transformation, derivation or computation rules of PM. A Godel statement can always be made for any system which can provide a Q function so it's quite powerful.

I've noted before, and note here again, that Truth of a contingent truth is different from proof by contradiction. To say that "birds cannot fly" is false because I can find a flying bird. But to say that "1 + 1 = 3" is false because "3=1+1+1" so "1+1 = 1+1+1" is different. "1+1=2" is then not true because I can find an instance of it, but because to accept anything else would upset our definitions and accepted system. Counting (and therefore maths) is based upon thousands of years of trading and accountancy culture - this is what is at stake. Inconsistent truth and contingent truth are different and I think this is the bit about logic that leaves a bad taste in the mouth. We can argue that "this person must have done the murder because there is no other way" but this is different from actually seeing them, or being murdered by them. There is always that possibility that out system is at fault. A friend said something at work which proves to be one of the wisest things I ever heard "Assumption is the mother of all cock-ups." When we get a contradiction the system doesn't decide it is true or false but rather we change our assumptions. Contradiction breaks systems and is undecidable for semantic rules. False however is defined by semantic systems. Contradictions aren't false, so Tautologies therefore aren't True. That is the point. Godel's statement isn't True then! Anyway the point of this blog is elsewhere... Before that: Semantic rules are simply rules that split the sentences of a logic into two piles (True and False usually). Tarski proved that these rules can't be part of the system because it creates a contradiction. So the "use" of a system must lie outside the system. I'm saying that actually there are 4 piles. True, False, Null (for contradictions) and whatever I decided to call the opposite of Null when I discussed this before. Anyway that sort of leads into the point. IN PM we can represent any number as 0SSSSS..S where S is the successor function. I presume we can derive any number from repeat application of thus formula.

So if there was a formula $(some number) then PM could derive any formula it wanted both nonsense and Godel statement! So the proof of Godel statement in PM would simply be: "$(0SSSSS.....S)" as many times as needed to get the Godel statament. Let us call this proof of Godel's statement "$(0G*S)" for G times S or simply X for chaos as well as big x. If "$(0G*S)" is the string which proves G then we have found an x. So X proves a contradiction. But it was obvious that the function $() could prove a contradiction cos it can prove nonsense as well like the statement "y))))Qf)))))". So it must be that there must be no formula in PM that can do what $() does on pain of death!

3/10/10: I have read the Godel paper now and indeed the Godel formula is not expressible in PM. Neither would the $() function above either. Firstly this must be true as argued above because it would enable the proof of any string, and secondly because there is no way in PM to handle characters. PM formulae take numbers as values not characters like "(". Godel goes to a lot of trouble in the paper to produce the function Bew which is the function which is true if the encoded sentence it is fed has a proof. But it relies on the "user" to do the encoding. I thought modern computer systems might be more powerful than PM but of course a modern computer (like the one running this blog page) works in numbers also. What we as users take to be characters of an alphabet are really ASCII codes which are converted into arrays of pixels for the screen. At no point in a computer is there a "character". Even BASIC functions like Mid$() don't return a character they return a number that is handled to produce a character. Characters are extrinsic to logical systems, they are meta the system!! So how do characters arise at all? Do characters even exist? Godel numbering looks simple but hides something tricky...

17/5/2022 note that numbers are abstract. Computers use switches! Numbers are encoded into switches.

investigate the relationship between characters and logic systems (do we go Searle here?)

SO that proves what I suspected at the beginning ages ago that Godel numbering is a necessary meta feature of a logical system. To talk of Godel numbering is already to step necessarily outside the system, so capping it by finding a sentence in the system which Godel numbering can talk about but which the system can't is really only a specific issue of something much bigger. For me the very fact that stepping outside a system is possible proves the rest already. Godel and "us" sit outside PM because we can Godel number. Who cares whether PM know this, we are already outside by necessity. I can't believe that Godel didn't realise that he was already outside PM - at least that Dangerous Ideas program seems to suggest he spent a lot of time trying to get outside.