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This construction is strictly stronger than GODEL's. It shows that for any sound system S, the implication "DEE runs implies DUM runs" is unprovable. The construction provides a proof of Godel's theorem, although it is similar to ROSSER (The statement DEE halts is probably NOT the negation of "DUM halts")

This construction is strictly stronger than GODEL's. It shows that for any sound system S, the implication "DEE runs implies DUM runs" is unprovable. The construction provides a proof of Godel's theorem, although it is similar to ROSSER (The statement DEE halts is provably NOT the negation of "DUM halts", that's the whole point)

This construction is strictly stronger than GODEL's. It shows that for any sound system S, the implication "DEE runs implies DUM runs" is unprovable. The construction provides a proof of Godel's theorem, although it is essentially equivalent to ROSSER. The statement DEE halts is the provable negation of "DUM halts", and the programs are saying that S proves "DEE halts" before "DUM halts"

This construction is strictly stronger than GODEL's. It shows that for any sound system S, the implication "DEE runs implies DUM runs" is unprovable. The construction provides a proof of Godel's theorem, although it is similar to ROSSER (The statement DEE halts is probably NOT the negation of "DUM halts")

TYPE I--- self referential pi-0-1 statements (statements about the non-halting of a certain computer program)

TYPE II --- these prove that there exist total functions which are not provably total. The statements in this case are pi-0-2, statements about the totality of some computable function.

TYPE III --- nonconstructive theorem about a large class of statements, which do not provide an explicit unprovable statement, and so cannot be used to step up the heirarchy of systems.

###TYPE I: self referential pi-0-1 statements (statements about the non-halting of a certain computer program)

###TYPE II: these prove that there exist total functions which are not provably total. The statements in this case are pi-0-2, statements about the totality of some computable function.

###TYPE III: nonconstructive theorem about a large class of statements, which do not provide an explicit unprovable statement, and so cannot be used to step up the heirarchy of systems.