Furthermore, this guy is asking for a "complete and consistant" definition.
In the Incompleteness Theorem, a system of axioms is complete if, for all statements in the system, either the statement or its negation is provable from the axioms. A system is consistant if there exist no statements for which both the statement and its negation are provable.
Basically, his "proof" is "Hey, we don't want a contradictory or unfinished definition, right? And those words mean the same thing as consistant and complete! So, Godel!"