The Relation Between G(?)del’s Incompleteness Theorem And Non-Euclidean Geometry

Non-Euclidean geometry and G(?)del’s incompleteness theorem were great mathematical achievements in nineteenth century and twentieth century respectively.This article mainly establishes the connection between the two through consistency.After the emergence of hyperbolic geometry,people doubted whether this new geometry was real,and mathematicians began to examine whether this new geometry was consistent.Mathematicians used the model method to reduce the consistency of non-Euclidean geometry to the consistency of Euclidean geometry.Then Hilbert established a more complete axiom system for Euclidean geometry in “The foundation of Geometry”,and at the same time defined a real number model for Euclidean geometry,reducing the consistency of Euclidean geometry to the consistency of real number arithmetic systems.With the efforts of mathematicians such as Dedekind,the consistency of the real number arithmetic system was reduced to the consistency of the natural number arithmetic system.This process could be continued,but these were indirect proofs of consistency,and no one could guarantee that the axiom system mentioned above will never produce contradictions.Therefore,it was necessary to directly prove the consistency of some axiom systems,so as to establish the consistency of other axiom systems.So,Hilbert proposed meta-mathematics to thoroughly solve the problem of consistency,hoping to directly prove the consistency of natural number arithmetic systems through limited meta-mathematical steps.In 1931,G(?)del proposed the incompleteness theorem and concluded that the consistency of natural number arithmetic systems could not be proved by Hilbert’s method.