What is Incompleteness

Kurt Gödel wrote the incompleteness theorems in the 1930s. They are still valid as the subject of several of today’s discussions and will continue to be effective in the future. These theorems show that any logical system consists of either contradiction or statements that cannot be proven. Therefore, they help to understand that the formal systems used are not complete. On the other hand, the trend of mathematicians in the 1900s was the unification of all theories for the solution of the most difficult problems in all disciplines. Before Gödel defined any system as consistent without any contradictions, and as incomplete with all or some disproved statements, Hilbert formulated all mathematics in an axiomatic form with Set Theory. Gödel’s theorem showed the limitations that exist within all logical systems and laid the foundation of modern computer science. These theorems caused several results about the limits of computational procedures. A famous example is the inability to solve the halting problem. A halting problem finds out whether a program with a given input will halt at some time or continue to run into an infinite loop. This decision problem demonstrates the limitations of programming. In a modern sense, this means that it is impossible to build an excellent compiler or a perfect antivirus.

In 1902, the logician Bertrand Russel disqualified the certitude given by Cantor’s proofs, presenting his paradox sets. Russel and Alfred Whitehead have developed a strong axiomatic system for framing types of mathematical reasoning in their system. These logical rules were trying to avoid the paradox. The system was, on the other side, very complicated. The mathematician, David Hilbert realized an easier system, trying to include all types of right judgment. He wanted to prove that this system is not contradictious, resulting that mathematics offers certainly knowledge. The essence of Hilbert’s schedule was shattered by the brilliant logician Kurt Gödel. This man, through his theorems of incompleteness, shows that any system with formal rules, large enough that it can contain arithmetical elements, has also unprovable sentences. Gödel’s incompleteness theorem shows that a theory that implies mathematical-axiomatic thinking, cannot be complete, meaning that it can have different internal limits or that cannot completely describe mathematically shaped physical reality. Gödel’s theorem is an expression on what structurally exists in the created reality of the world – meaning the limit. There is a correspondence between Gödel’s logical theorems and Heisenberg’s indetermination principle from quantum physics, by the fact that both express the inherent limitations of each axiomatical thinking or theory. Thereby, a scientific theory cannot pretend anymore that it’s the exclusive result of axiomatical thinking, in which intuition and spiritual experience have no role (Lemeni, 2009 AU14: The in-text citation "Lemeni, 2009" is not in the reference list. Please correct the citation, add the reference to the list, or delete the citation. , p. 141).