# Can you express a logic system like S5 using only a Gödel number?

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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It's not true that a logical system is just statements and axioms. There are also rules of inference. – Zhen Lin Aug 12 '11 at 2:59
Yes 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? – Programmer 400 Aug 12 '11 at 3:25
@Niklas: How about an uncountable language, which cannot be encoded by cardinality issues? – Asaf Karagila Aug 12 '11 at 4:46
@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. – André Nicolas Aug 12 '11 at 7:46
That's very interesting. Thank you for the insights. – Programmer 400 Aug 12 '11 at 18:16