FOM: comments on Wilson's dual view of foundations

[Stephen G Simpson]

This is a correction to my posting of yesterday on the same subject.

Harvey Friedman pointed out to me that my two statements

 > NF is interpretationally rich in the same sense and for the
 > same reasons as ZFC.

and 

 > It is known that topos theory + natural number object + additional
 > axioms is interpretationally relatively poor.

are hard to reconcile and perhaps misleading.  I need to clarify.

The actual facts are these.  It is known that topos + nat + additional
axioms interprets type theory, and vice versa.  So topos + nat +
additional axioms can interpret a lot of math but cannot interpret
ZFC.  It is known that NF interprets type theory, and it is open
whether NF interprets ZFC, or whether ZFC interprets NF.  It is
conceivable that NF is interpretationally extremely rich, as rich as
ZFC or even much richer, but we don't know this.

Summarizing these remarks and my earlier posting on this: Both NF and
topos + nat + additional axioms seem to fare poorly compared with ZFC
in terms of interpretational richness.  And they also seem to fare
poorly compared with ZFC in terms of an underlying foundational
picture.  These are the two aspects that Wilson discussed.

-- Steve