Generalizations and Strengthenings of Gödel’s Incompleteness Theorem

Abstract

In 1931 in the journal Monatshefte für Mathematik und Physik a short paper (a bit more than 20 pages) of an Austrian mathematician and logician Kurt Gödel was published — paper which has turned out to be one of the greatest and most important papers in mathematical logic and foundations of mathematics. Its title was “Über formal unentscheidbare Sätze der ‘Principia Mathematica’ und verwandter Systeme. I”. In it Gödel proved that arithmetic of natural numbers and all systems containing it are essentially incomplete provided they are consistent. It means that there are sentences which are undecidable in them, i.e. sentences φ such that neither φ, nor ¬φ are theorems. What’s more, we know which sentence of the pair φ, ¬φ is true in the basic model of the theory, i.e. in the model to the description of which the theory was formulated. This incompleteness is essential, i.e. it cannot be removed by adding the undecidable sentences as a new axioms because new undecidable sentences will appear (undecidable in the new, richer theory). This theorem (so called 1^st Gödel theorem) indicates the cognitive limitations of the deductive method.