Difficulties of the set of natural numbers

Difficulties of the set of natural numbers

I got this article recently:
http://arxiv.org/abs/1308.1018
Abstract says:
In this article some difficulties are deduced from the set of natural
numbers. By using the method of transfinite recursion we define an
iterative process which is designed to deduct all the non-greatest
elements of the set of natural numbers. But unexpectedly we meet some
difficulties in answering the question of whether the iterative process
can deduct all the elements of the set of natural numbers. The
demonstrated difficulties suggest that if we regard the class of natural
numbers as a set we will be confronted with either a semantic
contradiction or a conflict with the axiom of regularity. As a result, we
have the conclusion that the class of natural numbers is not a set but a
proper class.
He proves by means of a transfinite recursion that the greatest natural
number exists and that $\mathbb{N}$ contains itself as an element.
I am personally not a specialist for foundations of math, but it sounds
bravely and interesting for me.
What does the community think about this?