[FOM] Shelf life of inconsistent theories

[praatika@mappi.helsinki.fi]

> 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 
content)


Best, Panu

Panu Raatikainen

Ph.D., Academy Research Fellow,
Docent in Theoretical Philosophy

Department of Philosophy
University of Helsinki
Finland


E-mail: panu.raatikainen at helsinki.fi

http://www.mv.helsinki.fi/home/praatika/