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
Can you express a logic system like S5 using only a Gödel number?
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2It's not true that a logical system is just statements and axioms. There are also rules of inference. – 2011-08-12
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0Yes and no. Aren't they all just a Gödel number whether it's a statement, axiom or rules of inference? If we can write rules of inference with Gödel numbers then this way anything is just a Gödel number? Can you present a counterexample of anything in logic which doesn't have a Gödel number? – 2011-08-12
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2@Niklas: How about an uncountable language, which cannot be encoded by cardinality issues? – 2011-08-12
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1@Niklas R: Yes, you are completely right. Anything in $S_5$, for example the sentences and the derivations, can be assigned a natural number index in close analogy to the familiar Gödel numbering. But Gödel did his indexing with a definite purpose in mind, the Incompleteness Theorem. Any indexing for $S_5$ would, similarly, need to be done for a definite purpose. – 2011-08-12
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0That's very interesting. Thank you for the insights. – 2011-08-12
1 Answers
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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.