A Simplified Explanation of Gödel’s proof - Part 8
8: The Crux of the Proof
Page last updated 15 May 2021
Recall that in Part 4: Creating a special number relationship that if we can say:
‘Sequence A is the proof of sentence B’
then if NumberA is the matching Gödel number for sequence A, and NumberB is the matching Gödel number for sentence B, there’s a matching relationship which says:
‘NumberA is the G‑proof of NumberB’
Now, the relationship G‑proof is a number relationship, so that the expression ‘NumberA is the G‑proof of NumberB’ is also a number relationship, since it deals with nothing but numbers. If NumberB isn’t the Gödel number of any formal sentence, then obviously there can’t be a G‑proof of NumberB. But if it is the Gödel number of a formal sentence and it’s a valid formal sentence, then it’s possible that there could be a formal proof of that sentence, so it’s possible there could be a number that is the G‑proof of NumberB. That means that there can be a number relationship that says:
‘There is a number that is the G‑proof of NumberB’
Now, if NumberB is the Gödel number of a formal sentence B, either there is a proof of the formal sentence B or there isn’t. As long as we’ve defined it properly, then it’s obvious that the number relationship, ‘There is a number that is the G‑proof of NumberB’ has to be either true or false. If it’s false, then the converse is true, and the converse is ‘There is not a Gödel number that is the G‑proof of NumberB’. And really that’s the same as simply saying, ‘There is no G‑proof of NumberB’.
Recall that Part 5: Getting a number relationship to correspond to the concept of substitution was that a number relationship can be defined that precisely matches the concept of the substitution of a variable in a formal sentence by a particular number. We called it
G‑Substitution (Number1, Number2)
and which means the equivalent of substituting the free variable of a formal sentence by a number.
Gödel uses that concept of substitution. He takes a number relationship, and by a bit of clever manipulation which involves substituting the number relationship by its own matching Gödel number, he ends up with a new relationship which appears to refer to itself by its own Gödel number.
The first step in getting there is to define a relationship as:
‘There is no G‑proof of G‑Substitution (x, GN(x))’
This relationship isn’t a number relationship since it includes the Gödel numbering function GN(x). Because we want to end up with a number relationship that says essentially the same thing, we define another number relationship as:
‘There is no G‑proof of G‑Substitution (x, BN(x))’
Here we’ve used the Basic Numbering function BN instead of the Gödel numbering function GN. For short we call this relationship RS, so that RS(x) means the same as ‘There is no G‑proof of G‑Substitution (x, BN(x))’.
RS(x) is a number relationship, since it is defined only in terms of numbers and variables for numbers.
Now, since it’s a number relationship, there’ll be a corresponding formal sentence for this relationship. Since we called this relationship RS, I’m going to call this formal sentence FS. And for this formal sentence FS there’ll be a corresponding Gödel number, which I’m going to call GS.
At this point it can start to get confusing, so if you are at all unclear at this stage, you need to go back and read over it again. Remember that we have three matching items:
- the number relationship RS,
- the matching formal sentence FS,
- the matching Gödel number GS.
Each of these are all essentially referencing the same thing. You just have to remember that RS and FS both have a free variable, whereas GS, since it’s only a number, has no free variable.
Gödel’s trick at this stage is to substitute the free variable of the number relationship RS by the number GS. This is really the substitution of the free variable of the number relationship RS by its own matching Gödel number. That results in another number relationship, which is:
‘There is no G‑proof of G‑Substitution (GS, BN(GS))’.
For short we can call this number relationship RS(GS), so RS(GS) is equivalent to ‘There is no G‑proof of G‑Substitution (GS, BN(GS))’.
For this number relationship RS(GS), there is a corresponding formal language sentence, which we’ll call FS(GS). Now, as we said before, either there’s a proof in the formal language of this formal sentence FS(GS), or there isn’t.
We suppose for the moment that there a proof of FS(GS) in the formal language. In that case, the corresponding number relationship, which is RS(GS), and which means:
‘There is no G‑proof of G‑Substitution (GS, BN(GS))’ must be true.
Now if that really is true, then that means there can’t be any proof of the formal sentence where the free variable of the formal sentence FS is substituted by the number GS; that is the same as saying that there is no formal proof of FS(GS). But now we’ve got a contradiction. We started off supposing that there is a formal proof of FS(GS) and we end up with the conclusion that there isn’t a formal proof of the formal sentence FS(GS).
We arrived at a contradiction because we initially supposed that there is a formal proof of the formal sentence FS(GS), which is the sentence you get when you substitute the free variable of the sentence FS by the number GS. That means that supposition must be wrong. That means that there can’t be a formal proof of FS(GS).
Now, the number relationship that expresses the same concept as that formal sentence, the number relationship that is RS(GS) actually states:
‘There is no G‑proof of G‑Substitution (GS, BN(GS))’.
The only possible conclusion is that that number relationship is true (if the formal language system is consistent), but there cannot be any proof of that statement in the formal language.
And that completes our simplified version of Gödel’s proof of his incompleteness theorem.
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