HDS19: Doets, Kees (2016) Completeness and Definability. Doctoral thesis, University of Amsterdam.
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Abstract
Completeness and Definability
Applications of the Ehrenfeucht game in secondorder and intensional logic
Kees Doets
This dissertation contains results on classical first and secondorder logic (parts I and II) and their intensional colleagues: modal tense and intuitionistic (propositional) logic (part III).
One underlying theme is Ehrenfeucht's game and some of its variants
Chapter 1 is an Introduction to Ehrenfeucht game theorg and its relation with (quantifierrank) \alphaequivalence in (infinitary) logic. Section 1.0 intends to whet the appetite for the finite Ehrenfeucht game.
In chapter 2 the game is played on binary trees. A characterisation is obtained of all trees nequivalent with the binary tree B+m, all of whose branches have length m. In particular, it follows that B_m has _infinite_ nequivalents when m>=2^n1. This has been applied by Rodenburg [1986] to solve a problem in intuitionistic correspondence theory; the story is told in chapter 8.
Part II shows how to axiomatize certain monadic \Pi^1_1theories, most of them dealing with wellfoundedness. Chapter3 is on linear orderings. One of the nicer results is in 3.3 where the effects of the Suslin propertg for the monadic \Pi^1_1theory of H are isolated. Chapter 4 generalizes the method of 3 to the case of trees.
In part III, chapter 5 discusses LöwenheimSkolem type problems in modal correspondence theory. It is shown that most examples of nonfirstorder definable modal formulas already cannot be firstorder defined on _finite_ frames. On the other hand, an example is given of a nonfirstorder definable formula which _is_ firstorder definable on all countable frames.
Chapter 6 modifies the Ehrenfeucht game for use in intensional logic; exact Kripke models are constructed universal with respect to finite partially ordered Kripke models.
Chapter 7 presents our version of Zcompleteness.
In chapter 9, games and the universalexact Kripke model appropriate for onevariable intuitionistic formulas are applied to solve some problems in intuitionistic correspondence theory left open by Rodenburg [1952].
Appendix A constructs asymmetric linear orderings with lots of homogeneityproperties in each uncountable cardinal.
Appendix B reduces all of higherorder logic to monadic secondorder logic  indicating the expressive possibilities of modal logic in the Kripke semantics.
To help the reader find his way, here is an indication what can be omitted without loss of understanding of the rest. In chapter 1, sections 4, 8 and 9 are not needed for the other chapters. Also, not much will be lost if, in the discussion of the \alphagame, the reader always assumes \alpha to be _finite_. Section 2.4 can be read independently from the rest of chapter 2. Section 3.2 may be omitted. In part III, all chapters can be read individually (except for a couple of references where this is indicated.)
I am obliged to several people for different reasons; in particular I wish to thank here prof. Specker for a lecture featuring Ehrenfeucht games; Piet Rodenburg for the communication of his problems to which chapters 2/8/9 are devoted and the elimination of numerous mistakes in a previous version of this text; Anne Troelstra and Dick de Jongh for scientific support and the software used to produce this text on the Macintosh Plus.
However, above all, my gratitude concerns my thesisadvisor Johan van Benthem whose determination and persuasiveness eventually turned out to be irresistible.
Item Type:  Thesis (Doctoral) 

Report Nr:  HDS19 
Series Name:  ILLC Historical Dissertation (HDS) Series 
Year:  2016 
Additional Information:  Originally published: May 1987. 
Subjects:  Logic 
Depositing User:  Dr Marco Vervoort 
Date Deposited:  11 Jan 2022 23:20 
Last Modified:  11 Jan 2022 23:20 
URI:  https://eprints.illc.uva.nl/id/eprint/1851 
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