[FOM] Shelf life of inconsistent theories

Wayne Aitken waitken at csusm.edu
Mon Sep 17 23:48:57 EDT 2007

> Focus on finitely axiomatized theories (this is not a serious limitation).
> Assume then that for any inconsistent theory T with the length n, there is
> an upper limit m, which is in any case a recursive function of n, such
> that the proof of the inconsistency of T has at most the length m. This
> would provide us with a decision method for inconsistency. But we know
> well that there is no such decision method.

Here is an obvious way to create a system that is probably inconsistent,
but very hard to prove so: take ZFC + not P where P is a conjecture widely
believed to be true such as the Riemann hypothesis or the Goldbach
conjecture. It may be that a particular P is false or independent of ZFC.
However, it is very likely that at least some such P is provable in ZFC
with a proof that no one will be able to find for many decades or

On the other hand, it would be difficult to convince mathematicians to use
ZFC + not P as a foundation for mathematics.

I suppose that my hope (perhaps misplaced) is that foundational systems
are simple enough (in some sense) that any contradiction will likely
emerge relatively quickly.

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