Completeness of Logical and Mathematical Formal Systems: Types and Specifics

The article deals with the expansion of the concept of the completeness of a logical system. It is assumed that the generally accepted practice of correlating the concept of completeness primarily with the logic of the first order is due to purely historical circumstances. It is shown that when using first-order logic as a means of mathematical theorizing, the completeness of the logical system does not reflect the important features of the application of logic to mathematics. It is demonstrated that the distinction between deductive, semantic and descriptive completeness
leads to a new understanding of the role and nature of logic.

Keywords: logic, completeness, descriptive completeness, Gödel's theorem, mathematical theorizing.