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

Question

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