Is there Completeness in Mathematics after Gödel?

Abstract

What do mathematicians do? There are many ways of approaching this question, but one kind of answer is easy to give. Mathematicians study structures of different kinds: for instance, natural numbers, real numbers (the continuum), geometrical structures, groups, lattices, topological spaces, etc. For these structures, they develop corresponding theories, such as number theory, real analysis (theories of measure and integration), Euclidean geometry, group theory, lattice theory, topology, etc. In the late nineteenth century, a large and central class of foundational problems came up concerning the nature, the basic assumptions, and the presuppositions of theories of this general kind. These problems included questions concerning the axiomatization of important mathematical theories, the definitions of their basic concepts (e.g., natural numbers, real numbers, different geometrical objects), the relation of these mathematical theories to logic, etc. These questions are clearly among the foundational problems which working mathematicians are likely to find relevant. Hilbert’s efforts in his axiomatization of elementary geometry constitute a representative example of this early foundational work.1 KeywordsMathematical ThinkingLogical TruthStandard InterpretationUnderlying LogicNonstandard ModelThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.