arXiv:1203.0494cs.LO
Inconsistency of the Zermelo-Fraenkel set theory with the axiom of choice and its effects on the computational complexity
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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.
Although I am starting to get out of my depth, my impression was that ZFC itself is (per Godel's results) inconsistent or incomplete. Here is some research to this effect: If this is indeed the case (as it seems to my unprofessional first glance to be) then mathematics does not…
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