  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.
