[FOM] Shelf life of inconsistent theories
praatika at mappi.helsinki.fi
Mon Sep 17 13:50:08 EDT 2007
> 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.
In general, this can't be the case. Consider the following:
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.
(The qualification "of the type actually used in mathematics" might, of
course, make some difference, thought it is not easy to judge its exact
Ph.D., Academy Research Fellow,
Docent in Theoretical Philosophy
Department of Philosophy
University of Helsinki
E-mail: panu.raatikainen at helsinki.fi
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