Problems related to arithmetic

			Julia F. Knight

		   University of Notre Dame

I will describe two main groups of problems, giving a little background,
and indicating relationships among problems. 

The problems in the first group concern the difficulty in recovering, from a 
non-standard model A of first order Peano arithmetic (PA), the fragments of 
Th(A).

Much of what we know about models of arithmetic, and completions of PA, is
connected with Scott sets. The families of sets naturally coded in
completions of PA are exactly the countable Scott sets. The families of
sets coded in a model of PA is also a Scott set. It is still unknown,
without CH, whether all Scott sets arise this way. 

The problems in the second group concern the structure of Scott sets themselves.