The Uniformity Principle for Σ-definability

This article is an extended version of the paper published in Korovina and Kudinov (2007, Lecture Notes in Computer Science, Vol. 4497, pp. 416-425). The main goal of this research is to develop logical tools and techniques for effective reasoning about continuous data based on Sigma-definability. In this article we invent the Uniformity Principleand prove it for Sigma-definability over the real numbers extended by open predicates. Using the Uniformity Principle, we investigate different approaches to enrich the language of Sigma-formulas in such a way that simplifies reasoning about computable continuous data without enlarging the class of Sigma-definable sets. In order to do reasoning about computability of certain continuous data we have to pick up an appropriate language of a structure representing these continuous data. We formulate several major conditions how to do that in a right direction. We also employ the Uniformity Principleto argue that our logical approach is a good way for formalization of computable continuous data in logical terms.