[FOM] Shelf life of inconsistent theories

[Wayne Aitken]

Feng Ye said:

> time will silence the skeptics about consistency.

My intuition is that if a formal system (of the type actually used in
mathematics)
is inconsistent, then a contradiction will arise relatively quickly with
serious use.
So I would tend to agree with the above statement. However,  I know from
previous
comments on this forum that many people disagree. In fact, I understand that
some people would not be surprised if a contradiction were found in
Quine's NF.

There have been systems by Frege, Church, Curry, Quine, and others that have
turned out to be inconsistent. (I do not count Cantor since his set theory
was
not formalized before Zermelo and so had some wiggle room).

My question is what is the longest period of time that an inconsistent
system has been in
active use before a contradiction was discovered? Here "active use" is a
bit vague,
but ideally this means several mathematicians besides the originator were
using
it to prove serious mathematical theorems. I include in this any proposed
large
cardinal axiom in set theory that has later turned out to contradict ZF or
even ZFC.