A-saturated 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 A-saturated, 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 ∆n-definable elements of M. In Chapter 3 we look at the substructure of M determined by its set of ∑n-definable 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 π1-consequences of PA determined by an element of K and show that in certain circumstances these initial segments are A-saturated; 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 A-saturated structures introduced in Chapter 4, showing the existence of approximating chains of models of PA for certain models of the set of π2-consequences 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.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:490556 |
| Date | January 1978 |
| Creators | Lessan, Hamid |
| Publisher | University of Manchester |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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