Number Theoretic Density and Logical Limit Laws

American Mathematical Soc., Jan 1, 2001 - Mathematics - 289 pages
This book shows how a study of generating series (power series in the additive case and Dirichlet series in the multiplicative case), combined with structure theorems for the finite models of a sentence, lead to general and powerful results on limit laws, including $0 - 1$ laws. The book is unique in its approach to giving a combined treatment of topics from additive as well as from multiplicative number theory, in the setting of abstract number systems, emphasizing the remarkable parallels in the two subjects. Much evidence is collected to support the thesis that local results in additive systems lift to global results in multiplicative systems. All necessary material is given to understand thoroughly the method of Compton for proving logical limit laws, including a full treatment of Ehrenfeucht-Fraisse games, the Feferman-Vaught Theorem, and Skolem's quantifier elimination for finite Boolean algebras. An intriguing aspect of the book is to see so many interesting tools from elementary mathematics pull together to answer the question: What is the probability that a randomly chosen structure has a given property? Prerequisites are undergraduate analysis and some exposure to abstract systems.

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Contents

 Counting Functions and Fundamental Identities 17 Density and Partition Sets 45 The Case p 1 75 The Case 0 p 1 87 Monadic SecondOrder Limit Laws 103 Background from Analysis 127 Counting Functions and Fundamental Identities 143 Chapter g Density and Partition Sets 159
 The Case 0 a oc 201 FirstOrder Limit Laws 217 Appendix A Formal Power Series 233 Appendix B Refined Counting 251 Consequences of 6P 0 261 On the Monotonicity of an When pn 1 269 Symbol Index 285 Copyright

 The Case a 0 195