[FOM] Finite axiomatisation
Stephen G Simpson
simpson at math.psu.edu
Thu Aug 21 09:43:49 EDT 2008
Alasdair Urquhart writes:
>
> > Also, there is some independent interest, I would think, in knowing
> > the shortest, or close to the shortest, finite axiomatisation of PA.
> >
> > Does anyone have any ideas on how it would look? Does anyone
> > want to try it?
>
> Perhaps the simplest way would be to apply the same trick that
> generates NBG set theory from ZFC, that is to say, add a set of
> axioms defining first-order properties. It could be that this
> has been already done in the literature.
Yes, this is very much in the literature. See for instance my book
"Subsystems of Second Order Arithmetic," where the significance of
ACA_0 in reverse mathematics and foundations of mathematics generally
is discussed. There it is pointed out that ACA_0 is a finitely
axiomatizable conservative extension of PA, analogously to how NBGC is
a finitely axiomatizable conservative extension of ZFC. I have not
tried to write a short axiomatization of ACA_0, but clearly this could
be done along the lines suggested by Alasdair.
-- Steve
Name: Stephen G. Simpson
Affiliation: Professor of Mathematics, Pennsylvania State University
Research Interests: mathematical logic, foundations of mathematics
Web page: http://www.math.psu.edu/simpson/.
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