Real set theory (Q1972505)

The author defines a ternary Ackermann-like arithmetic function on the natural numbers and declares it to be valid for cardinal numbers. Transforming his declaration into an axiom (the Axiom of Monotonicity), he arrives at an extension of ZF that he calls Real Set Theory (RST). No consideration is given to the consistency of RST. It is claimed that the GCH is a theorem of RST and hence so is AC. Evidently feeling the need to extend the underlying logical basis for ZF, the author goes on to add three new derivation rules to the predicate calculus, arriving at the \textit{Enhanced Predicate Calculus}. For example, his \textit{Contradiction Rule} says that any formula leading to a contradiction cannot be derived. Using his derivation rules the author claims that he can prove ``first and second incompleteness theorems'' for RST without using any metalanguage. Why these results are labelled ``incompleteness theorems'' is difficult to discern.