FOM: modeling PRA in the physical world

[Stephen G Simpson]

Raatikainen Panu A K 16 Mar 1999 14:01:45 writes:

 > My point about Robinson Arithmetic was that your eariler claim ("
 > nested quantification over a domain ... requires an ontological
 > commitment to the (actual) existence of that domain") is problematic -

Well, I want to retreat from that claim.  What I really intended to
say was: In order to justify induction with respect to properties
defined by nested quantification over a domain, we seem to need an
ontological commitment to the actual existence of that domain, as a
completed totality.  The thought behind this is that induction only
makes sense for `definite' properties, and how can we expect
quantification over an incomplete totality to give rise to a
`definite' property?

Robinson's arithmetic is a good example of a theory where you have
quantification over N but you are not commiting yourself to the actual
existence of N as a completed totality.  Another example is PRA
formulated in predicate calculus with quantifiers over N -- this is
conservative over the quantifier-free version of PRA.

-- Steve