End Rosser's_Incompleteness.
Definition RepresentsInSelf (T:System) :=
exists rep:Formula, exists v:nat,
(forall x : nat, In x (freeVarFormula LNN rep) -> x = v)  /\
(forall f : Formula,
mem Formula T f ->
SysPrf T (substituteFormula LNN rep v (natToTerm (codeFormula f)))) /\
(forall f : Formula,
~ mem Formula T f ->
SysPrf T
(notH (substituteFormula LNN rep v (natToTerm (codeFormula f))))).
Definition DecidableSet (A:_)(s:Ensemble A) :=
(forall x : A,
mem A s x \/
~ mem A s x).

Theorem Incompleteness :
forall T : System,
Included Formula NN T ->
RepresentsInSelf T ->
DecidableSet Formula T ->
exists f : Formula,
(Sentence f) /\
(SysPrf T f \/ SysPrf T (notH f) -> Inconsistent LNN T).