[FOM] Meta-Logic

[praatika@mappi.helsinki.fi]

Paul Studtmann wrote:

> > I am interested in knowing what the weakest systems are that
> > can prove basic meta-logical theorems.  For instance, I would
> > like to know whether Peano Arithmetic can prove soundness and
> > completeness for first order predicate calculus.  Can anyone
> > either state such results or direct me to the relevant
> > literature?

A weak subsystem of second-order arithmetic called ACA_0, which is a 
conservative extension of Peano Arithmetic, is sufficient for all basic 
meta-logical results, including completeness theorem. This theory has been 
studied a lot in Reverse Mathematics:
http://en.wikipedia.org/wiki/Reverse_mathematics

(Roger Bishop Jones suggests that Robinson Arithmetic Q would suffice. It 
does not.) 

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