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This proceedings volume contains most of the invited talks presented at the colloquium. The main topics treated are the model theory of arithmetic and algebra, the semantics of natural languages, and applications of mathematical logic to complexity theory. The volume contains both surveys by acknowledged experts and original research papers presenting advances in these disciplines.

Mathematical logic

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The first part of this book concerns the present state of the theory of chains (= total or linear orderings), in connection with some refinements of Ramsey's theorem, due to Galvin and Nash-Williams. This leads to the fundamental Laver's embeddability theorem for scattered chains, using Nash-Williams' better quasi-orderings, barriers and forerunning. The second part (chapters 9 to 12) extends to general relations the main notions and results from order-type theory. An important connection appears with permutation theory (Cameron, Pouzet, Livingstone and Wagner) and with logics (existence criterion of Pouzet-Vaught for saturated relations). The notion of bound of a relation (due to the author) leads to important calculus of thresholds by Frasnay, Hodges, Lachlan and Shelah. The redaction systematically goes back to set-theoretic axioms and precise definitions (such as Tarski's definition for finite sets), so that for each statement it is mentioned either that ZF axioms suffice, or what other axioms are needed (choice, continuum, dependent choice, ultrafilter axiom, etc.).

Mathematical logic

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#gsdbf --- Philosophy of nature --- Mathematical logic

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Mathematical logic --- 51 --- Mathematics --- Model theory. --- Set theory. --- 51 Mathematics

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This book gives a complete and elementary account of fundamental results on hyperfinite measures and their application to stochastic processes, including the *-finite Stieltjes sum approximation of martingale integrals. Many detailed examples, not found in the literature, are included. It begins with a brief chapter on tools from logic and infinitesimal (or non-standard) analysis so that the material is accessible to beginning graduate students.

Mathematical logic --- Measure theory. Mathematical integration --- Stochastic processes