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I have 3 questions and they are closely related so I asked them in same post.

  1. With Gödel numbering we can encode statements like "0 = 0" or maybe "a then b". And this is basically just transformation of symbols or group of symbols to a unique number. My question is how do we encode This statement is cannot be proved from the axioms (which bring us to Gödel's Incompleteness Theorem) or any other statement that is not written with mathematical symbols but with words in any language?

  2. Other than that, what are the limits of Gödel numbering? I mean is it possible to encode "I like to play violin"?

  3. Does that mean if we can write any statement with Gödel number, it can be also written with mathematical equation?

Probably I have missed something or maybe the resources that I am trying to learn are not enough which lead me to these question. So if you have any resource that maybe helpful to understand topic better and different than these(where I learned):

https://www.youtube.com/watch?v=O4ndIDcDSGc

https://www.youtube.com/watch?v=HeQX2HjkcNo

it would be really great.

Muber17
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  • "how do we encode any statement that is not written with mathematical symbols but with words in any language?" Gödel encoding technique applies to formalized languages. – Mauro ALLEGRANZA Sep 06 '22 at 09:30
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    "is it possible to encode "I like to play violin"?" In principle yes: use enough prime numbers to encode all letters( plus dot, comma, question mark) of the alphabet. But the result will be quite useless. – Mauro ALLEGRANZA Sep 06 '22 at 09:32
  • See here and here for some examples in encoding. – Mauro ALLEGRANZA Sep 06 '22 at 09:33
  • "Does that mean that we can write any statement with mathematical equation?" The Gödel encoding produces for every formula of the formalized language a number and not an equation. – Mauro ALLEGRANZA Sep 06 '22 at 09:41
  • @MauroALLEGRANZA You are saying Gödel encoding is producing a number not an equation. Then it is just transformation, which means it is not representing anything. Like I am giving numbers to alphabet with a rule that I determine. If this is the case then how do we relate "This statement is cannot be proved from the axioms" statement's Gödel encoded version with mathematics. It is just cryptography. Where is my mistake? – Muber17 Sep 06 '22 at 10:58

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