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

[Xavier Noria]

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?

-- fxn