Each effectively generated system has its own Gödel sentence. It is possible to define a larger system that contains the whole of plus as an additional axiom. This will not result in a complete system, because Gödel's theorem will also apply to , and thus also cannot be complete. In this case, is indeed a theorem in , because it is an axiom. Because states only that it is not provable in , no contradiction is presented by its provability within . However, because the incompleteness theorem applies to , there will be a new Gödel statement for , showing that is also incomplete. will differ from in that will refer to , rather than .
Each effectively generated system has its own Gödel sentence. It is possible to define a larger system that contains the whole of plus as an additional axiom. This will not resu,lt in a complete system, because Gödel's theorem will also apply to , and thus also cannot be complete. In this case, is indeed a theorem in  because it is an axiom. Because states only that it is not provable in , no contradiction is presented by its provability within . However, because the incompleteness theorem applies to , there will be a new Gödel statement for , showing that is also incomplete. will differ from in tha,t will refer to  rather than .