Deductive Systems and Incompleteness

In antiquity, scientific discourse developed rapidly. Arithmetic and geometric theorems were precisely derived under the assumption of axioms. Euclid’s influential book “Elements” is an expression of this culture. From today’s perspective, every mathematical proof is based on a syntacticSyntax calculus, which we refer to as a deductive systemDeductive system. This concept includes a formal languageFormal language and logical inference rules. After further developments of AristotelianAristotle logic by LlullLlull, Ramón, LeibnizLeibniz, Gottfried Wilhelm, BolzanoBolzano, Bernard and others, it was only FregeFrege, Gottlob in his “Begriffsschrift” who introduced a deductive systemDeductive system of today’s kind. At the same time, recursion theory and the axiomatics of arithmetic were developed by DedekindDedekind, Richard. Shortly afterwards, PeanoPeano, Giuseppe laid the foundations of today’s notation in logic.