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Since logic systems are just statements and/or axioms, can we formulate a logic system gödel numbering the system itself so that the system becomes nothing but a gödel number? For instance the modal system S5 would become just a Gödel number? Is it true that any logical statement must have a Gödel number and are there statements which don't have a Gödel number? Thank you in advance

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    That's very interesting. Thank you for the insights.2011-08-12

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You can assign Gödel numbers to formulae in modal logic but it will not necessarily be interesting.

For one thing, you won't be able to refer to those numbers within the modal system so you don't get self-reference.

Also, $S_5$ is a decidable theory, whereas Gödel essentially used Gödel numbers to show that Peano arithmetic is not decidable.