A PROOF OF COMPLETENESS FOR CONTINUOUS FIRST-ORDER LOGIC

作者:Ben Yaacov Itai*; Pedersen Arthur Paul
来源:Journal of Symbolic Logic, 2010, 75(1): 168-190.

摘要

Continuous first-order logic has found interest among model theorists who wish to extend the classical analysis of "algebraic" structures (such as fields, group. and graphs) to various natural classes of complete metric structures (such as probability algebras, Hilbert spaces. and Banach spaces). With research in continuous first-order logic preoccupied with studying the model theory of this framework, we find a natural question calls for attention. Is there an interesting set of axioms yielding a completeness result?
The primary purpose of this article is to show that a certain. interesting set of axioms does indeed yield a completeness result for continuous first-order logic. In particular. we show that in continuous first-order logic a set of formulae is (completely) satisfiable if (and only if) it is consistent. From this result it follows that continuous first-order logic also satisfies an approximated form of strong completeness, whereby Sigma satisfies phi (if and) only if Sigma proves phi - 2(-n) for all n < omega. This approximated form of strong completeness asserts that if Sigma satisfies phi, then proofs from 1, being finite. can provide arbitrarily better approximations or the truth of phi.
Additionally, we consider a different kind of question traditionally arising in model theory-that of decidability. When is the set of all consequences of a theory (in a countable. recursive language) recursive? Say that a complete theory T is decidable if for every sentence W, the value phi(T) is a recursive real, and moreover, uniformly computable from p. If T is incomplete. we say it is decidable if for every sentence phi the real number phi(T)degrees is uniformly recursive from V, where phi(T)degrees is the maximal value of phi consistent with T. As in classical first-order logic, it follows from the completeness theorem ofcontinuous first-order logic that if a complete theory admits a recursive (or even recursively enumerable) axiomatization then it is decidable.

  • 出版日期2010-3