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Copyright   James R Meyer    2012 - 2024 https://www.jamesrmeyer.com

A Simplified Explanation of Gödel’s Incompleteness Proof:
Part 6




Matching number relationships and formal sentences

Page last updated 15 May 2021


We now get to the point in the proof where we assert:


For any number relationship with one free variable, there is a matching formal sentence with one free variable, where that formal sentence expresses the same concept as that number relationship.


NB: At this point we have to remark that in the above, and in the rest of the following argument, when we use the term number relationship, we mean number relationships that aren’t formal sentences. (Footnote: This requirement is also implicit in Gödel’s actual paper, though he never explicitly states it. His proof requires the notion of two totally separate types of number-theoretic relationships, even though the formal sentences can satisfy the definition of a number-theoretic relationship:
(1) number relationships that aren’t expressions of the formal language and
(2) expressions of the formal language. )


A relationship that’s a purely number relationship is simply a relationship between numbers that is expressed in some language that is not the formal language. For example, there can be many different ways of expressing the concept of x plus x is equal to six’. We could say x and x add up to six’, or we could say ‘two times x adds up to six’. But as long as it is clearly defined, then it could always be translated into the formal language.


And we know that for any formal sentence, there’s a matching Gödel number.


That means that, given a number relationship with one free variable, since there’s a matching formal sentence with one free variable, and since that formal sentence has a matching Gödel number, that number is also a matching Gödel number for that number relationship.


As a simple example, we take the number relationship x plus x is equal to six’. Let’s assume that in our formal language the matching formal sentence is:


{x + x = ffffff 0}


where in the formal system 0 is zero, f 0 is one, ff 0 is two, fff 0 is three, and so on.


Purely for the purposes of demonstration we assume that the matching Gödel number for that formal sentence is 12743 (although it would actually be a much bigger number). That means the matching Gödel number for x plus x is equal to six’ is also 12743.


Now, a number relationship with a free variable isn’t a proposition. But if the variable is substituted by some specific value, it can be a proposition. So if we substitute the free variable of our number relationship by some number, and we substitute the free variable of the formal sentence by the same number, then we can have a number relationship that is a proposition about numbers that has a matching formal sentence that is a sentence about numbers.


For our example, if we substituted the x by three, we would get ‘three plus three is equal to six’, and the matching formal sentence would be {fff 0 + fff 0 = ffffff 0}.


Provability and truth

Now that we have a matching formal sentence and a number relationship, then we can say that:


‘If a formal sentence is provable, then the matching number relationship must be true.’


It’s pretty obvious that that has to be the case – provided that the formal language is consistent (that means it isn’t able to prove false sentences).


So, with our previous example, if the formal language can prove that {fff 0 + fff 0 = ffffff 0}, then it has to be true that three plus three is equal to six.


And following on from the above, we can now say that:


‘For any sentence of the formal language, either there’s a formal proof of that sentence or there isn’t.’







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Copyright   James R Meyer   2012 - 2024