I am not sure what the reference is to - there should be no disagreeing with Godel's theorem of incompleteness - they were proved mathematically by Godel in 1931 by basically showing a mapping of any problem space to different numerical spaces and proving that there is always a rule that can not be proved by existing axioms. The theorem is proved conclusively mathematically, people that do not agree with it simply do not grasp the math. Godel used a standard method of "proof of impossibility" - the only issue is that it is a really complicated method to get there - but it is proven and have been verified many times over by following the math.
(*) Proof of impossibility requires showing one example that breaks a rule to render prove it impossible to enforce. Godel showed a way to map any closed system to an irrational number space and showed that the order of groups within irrational numbers proves that there is always that instance that breaks the rule. (You will have to excuse me for trying to simplify it - it was more than 20 years since I tried to go through the paper and got to a point where I actually understood the basic mathematical concepts in it).
People can disagree with how Godel later tried to describe his personal belief in god etc... - but the theorem itself is proven mathematically.