[FOM] extensions of PA that prove "false" statements

[praatika@mappi.helsinki.fi]

Lainaus Xavier Noria <fxn at hashref.com>:

> I read in Torkel Franzén's book on Gödel's Theorem that there are  
> consistent extensions of PA which prove statements which are  
> mathematically "false" (I guess given some natural interpretation of  
> such extensions).
> 
> I am curious about whether they are just a technicality, does anybody  
> have a pointer to some example?

Let Cons(PA) be the arithmetical formula expressing the statement that PA 
is consistent. By Gödel's second theorem, neither Cons(PA) nor its 
negation is provable in PA. Consequently, one can consistently add 
not-Cons(PA) to PA. The result is a consistent extension of PA which 
proves a (presumably) false statement, namely, that PA is inconsistent. 

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.helsinki.fi/collegium/eng/Raatikainen/raatikainen.htm