Kurt Gödel (1906-1978) was an Austrian mathematician who did pioneering work in logic and the foundations of mathematics. His Incompleteness Theorem demonstrated some limitations of the program that would have placed all of mathematics on a complete axiomatic basis. He immigrated to the United States and worked at the Institute for Advanced Study at Princeton, New Jersey.
|“||How Gödel transformed set theory can be broadly cast as follows: On the larger stage, from the time of Cantor, sets began making their way into topology, algebra, and analysis so that by the time of Gödel, they were fairly entrenched in the structure and language of mathematics. But how were sets viewed among set theorists, those investigating sets as such? Before Gödel, the main concerns were what sets are and how sets and their axioms can serve as a reductive basis for mathematics. Even today, those preoccupied with ontology, questions of mathematical existence, focus mostly upon the set theory of the early period. After Gödel, the main concerns became what sets do and how set theory is to advance as an autonomous field of mathematics.||”|
Gödel published his remarkable proof of the Incompleteness Theorem in 1931. He showed that in any consistent (first-order) axiomatic mathematical system there are always propositions that cannot be proved or disproved using the axioms of the system. He additionally showed that it is impossible to prove the consistency of the axioms from those same axioms. This was the famous incompleteness theorem: any axiomatic system powerful enough to describe arithmetic on natural numbers cannot be both consistent and complete. Moreover, the consistency of the axioms cannot be proven within the system.
Gödel's work abruptly ended a half-century of attempts, beginning with the work of Frege and culminating in Principia Mathematica and Hilbert's formalism, to find a set of first-order axioms for all of mathematics that is both provably consistent as well as complete. Bertrand Russell had already published, in Principia Mathematica (1910–13), a massive attempt to axiomatize mathematics in a consistent way. Gödel's proof also showed that the formalist approach of David Hilbert was bound to fail to prove consistency.
The incompleteness theorems also imply that there is no mechanical procedure which would determine, for all sentences of mathematics S, whether or not S was a theorem of the axioms for mathematics.
Gödel's proof was a landmark for mathematics, and demonstrated that it can never be a finished project.
Among Gödel's other remarkable achievements: the first to discover a solution to the equation for general relativity in which there are closed, time-like curves. This means it is mathematically possible for there to be universes in which one can go back in time (provided one has enough fuel and time—something probably not physically possible). Gödel was also the first to recognize the significance of the P=NP problem, in a letter he wrote to John von Neumann in 1956.
A profoundly religious man who reportedly read the Bible every morning, Gödel is also noted for giving Gödel's Ontological Proof, an attempt to make Anselm's ontological argument into a completely logically rigorous argument. This had the useful property of making very explicit and precise the assumptions necessary for one to accept the ontological argument. In this proof, Gödel demonstrated that if one accepts only five modest and seemingly obvious axioms, it is necessary to conclude that God exists.
- A paper about Godel's ontological proof of the existence of God. Godel did not publish his proof of the existence of God until 1971.