I was happily surfing the arXiv, when I was jolted by the following paper:

[Inconsistency of the Zermelo-Fraenkel set theory with the axiom of choice and its effects on the computational complexity][1] by *M. Kim*, Mar. 2012.

> **Abstract.** This paper exposes a contradiction in the Zermelo-Fraenkel set theory with the axiom of choice (ZFC). While Godel's incompleteness theorems state that a consistent system cannot prove its consistency, they do not eliminate proofs using a stronger system or methods that are outside the scope of the system. The paper shows that the cardinalities of infinite sets are uncontrollable and contradictory. The paper then states that Peano arithmetic, or first-order arithmetic, is inconsistent if all of the axioms and axiom schema assumed in the ZFC system are taken as being true, showing that ZFC is inconsistent. The paper then exposes some consequences that are in the scope of the computational complexity theory.

Now this seems to be a very major claim, and I lack the background to be able to judge if the claim is true, or there is some subtle or even obvious defect in the paper's arguments. But picking on this paper itself is **not** the purpose of my question.

**1.** The paper, however, got me very curious about how "disastrous" would inconsistency of ZFC really be? 

**2.** A slightly more precise question is: what would be the major consequences of the kind of inconsistency claimed in the abstract cited above?

If you feel that my questions might not admit "clearly right" answers, I will be happy to make this post CW. 


  [1]: http://arxiv.org/abs/1203.0494