As first proved by Ehrenfeucht and Mostowski [EM], every first-order theory which has infinite models, has models with infinite sets of indiscernibles. Ramsey's Theorem is a crucial component of the proof of this result. If the structure has built-in Skolem functions, taking the Skolem hull of the set of indiscernibles will produce structures generated by a set of indiscernibles.
In this thesis I study the question: Which first-order structures are generated by indiscernibles? J. Schmerl showed that if [special characters omitted] is a finite language, every countable recursively saturated [special characters omitted]-structure in which a form of coding of finite functions is available is generated by indiscernibles. Further, he showed that such a structure has arbitrarily large extensions which are generated by a set of indiscernibles, resplendent, and [special characters omitted]-equivalent to the original structure. Proofs of these theorems are complex and use a combinatorial lemma whose proof in Schmerl's paper has an acknowledged gap. I offer a complete proof of a more direct combinatorial lemma from which Schmerl's theorems follow.
The other subject of this thesis is cofinal extensions of linearly ordered structures. It is related to the work of R. Kaye who used a weak notion of saturation to give a sufficient condition under which a countable model of PA− has a proper elementary cofinal extension. I give two different proofs of the fact that every countable recursively saturated linearly ordered structure with no last element has a proper cofinal elementary extension.
|Commitee:||Kossak, Roman, Miller, Russell, Rothmaler, Philipp, Schoutens, Hans|
|School:||City University of New York|
|School Location:||United States -- New York|
|Source:||DAI-B 74/06(E), Dissertation Abstracts International|
|Keywords:||Cofinal extension, Indiscernibles, Peano arithmetic, Ramsey's theorem, Recursive saturation, Resplendence|
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