## Model Theory in Admissible Sets: The Theory of the Snake |

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### Contents

INTRODUCTION TO MODEL THEORY | 29 |

SOME BASIC RESULTS OF MODEL THEORY | 46 |

SOME APPLICATIONS OF SCOTT SENTENCES | 80 |

### Common terms and phrases

A-characterizable A-finite model A-finite structure A-finiteness lemma A-language A-recursive function admissible ordinal arbitrary admissible sets automorphisms axiom of extensionality Barwise Compactness theorem canonical Scott sentence cardinality CHAPTER class of formulas clearly Conjecture Consequently constant symbols continuum countable admissible set countable structures define definition dense order end extension F is A-recursive fact finite follows free variables function with domain H(co homogeneous linear ordering homogeneous models I-model induction hypothesis infinitary infinitary logic invariably characterizable k-tuple of elements Kunen language least element Lowenheim-Skolem property LOWENHEIM-SKOLEM RESULTS Lowenheim-Skolem theorem m,p e mapping model theory notion order type p-th block partial Scott formula potentially isomorphic power set power set axiom proof property iff pseudo-well-ordering quantifier rank quasi-A-f quasi-A-finite structure recursively inaccessible relation set theory subset Suppose theory on admissible thesis transitive set well-ordered Z-formula