Asaturated and recursively saturated structures were
introduced and extensively studied by Wilmers [Wi] and
Barwise  Schlipf [B & S] respectively. These structures
appear naturally in the study of models of arithmetic and
their special properties help us to solve some problems
regarding these models. Let PA denote the set of Peano's
axioms and Ma model of PA.
In Chapter 2 we show the existence of end extensions of M
which are Asaturated, using this we prove some results
regarding the initial segments of M and answer a question
of McAloon concerning the position of the class of
∆ndefinable elements of M.
In Chapter 3 we look at the substructure of M determined
by its set of ∑ndefinable elements and the initial segment
of M determined by this set. It turns out that these structures
are models of substantial parts of PA and that the set of
standard integers is definable in them. In the second part
of this Chapter we give a 'partial' solution to a problem of
Gaifman concerning arithmetical structures.
In Chapter 4 we consider the initial segments of a model
K of the set of π1consequences of PA determined by an
element of K and show that in certain circumstances these
initial segments are Asaturated; using this we generalize
some results of Chapter 2 to models of weaker systems than PA.
In Chapter 5 we continue to exploit the versatility of
the Asaturated structures introduced in Chapter 4,
showing the existence of approximating chains of models
of PA for certain models of the set of π2consequences
of PA. In the last section of this Chapter we introduce
a useful class of initial segments of M and prove some results
concerning the number of isomorphic initial segments of an
initial segment I of M and its relation to the number of
automorphisms of I, when M is countable.
Certain results concerning the relative position of the
classes of ∑n definable elements of a model of PA can
be found in 3.1 . and 4.2.
