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Godel's Incompleteness Theorem - in a nutshell.

“Anything you can draw a circle around cannot explain itself without referring to something outside the circle – something you have to assume but cannot prove.”

Kurt Friedrich Gödel (/ˈkɜrt ɡɜrdəl/; German: [ˈkʊʁt ˈɡøːdəl]; April 28, 1906 – January 14, 1978) was an Austrian, and later American, logician, mathematician, and philosopher. Considered with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when others such as Bertrand Russell, A. N. Whitehead, and David Hilbert were pioneering the use of logic and set theory to understand the foundations of mathematics.

Gödel published his two incompleteness theorems in 1931 when he was 25 years old, one year after finishing his doctorate at the University of Vienna. The first incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (for example Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms. To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers.

He also showed that neither the axiom of choice nor the continuum hypothesis can be disproved from the accepted axioms of set theory, assuming these axioms are consistent. The former result opened the door for mathematicians to assume the axiom of choice in their proofs. He also made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, and modal logic.

World War II started in September 1939. Before the year was up, Gödel and his wife left Vienna for Princeton.

Albert Einstein was also living at Princeton during this time. Gödel and Einstein subsequently developed a strong friendship, and were known to take long walks together to and from the Institute for Advanced Study. The nature of their conversations was a mystery to the other Institute members. Economist Oskar Morgenstern recounts that toward the end of his life Einstein confided that his "own work no longer meant much, that he came to the Institute merely ... to have the privilege of walking home with Gödel".

On December 5, 1947, Einstein and Morgenstern accompanied Gödel to his U.S. citizenship exam, where they acted as witnesses. Gödel had confided in them that he had discovered an inconsistency in the U.S. Constitution, one that would allow the U.S. to become a dictatorship. Einstein and Morgenstern were concerned that their friend's unpredictable behavior might jeopardize his chances. Fortunately, the judge turned out to be Phillip Forman. Forman knew Einstein and had administered the oath at Einstein's own citizenship hearing. Everything went smoothly until Forman happened to ask Gödel if he thought a dictatorship like the Nazi regime could happen in the U.S. Gödel then started to explain his discovery to Forman. Forman understood what was going on, cut Gödel off, and moved the hearing on to other questions and a routine conclusion.

During his many years at the Institute, Gödel's interests turned to philosophy and physics. In 1949, he demonstrated the existence of solutions involving closed time-like curves, to Albert Einstein's field equations in general relativity. He is said to have given this elaboration to Einstein as a present for his 70th birthday. His "rotating universes" would allow time travel to the past and caused Einstein to have doubts about his own theory. His solutions are known as the Gödel metric (an exact solution of the Einstein field equation).

He studied and admired the works of Gottfried Leibniz, but came to believe that a hostile conspiracy had caused some of Leibniz's works to be suppressed. To a lesser extent he studied Immanuel Kant and Edmund Husserl. In the early 1970s, Gödel circulated among his friends an elaboration of Leibniz's version of Anselm of Canterbury's ontological proof of God's existence. This is now known as Gödel's ontological proof. Gödel was awarded (with Julian Schwinger) the first Albert Einstein Award in 1951, and was also awarded the National Medal of Science, in 1974.

In later life, Gödel suffered periods of mental instability and illness. He had an obsessive fear of being poisoned; he would eat only food that his wife, Adele, prepared for him. Late in 1977, she was hospitalized for six months and could no longer prepare her husband's food. In her absence, he refused to eat, eventually starving to death.  He weighed 65 pounds (approximately 30 kg) when he died. His death certificate reported that he died of "malnutrition and inanition caused by personality disturbance" in Princeton Hospital on January 14, 1978. Adele's death followed in 1981.

Religious views

Gödel was a convinced theist. He held the notion that God was personal, which differed from the religious views of his friend Albert Einstein.

He believed firmly in an afterlife, stating: "Of course this supposes that there are many relationships which today's science and received wisdom haven't any inkling of. But I am convinced of this [the afterlife], independently of any theology." It is "possible today to perceive, by pure reasoning" that it "is entirely consistent with known facts." "If the world is rationally constructed and has meaning, then there must be such a thing [as an afterlife]."

In an unmailed answer to a questionnaire, Gödel described his religion as "baptized Lutheran (but not member of any religious congregation). My belief is theistic, not pantheistic, following Leibniz rather than Spinoza." Describing religion(s) in general, Gödel said: "Religions are, for the most part, bad—but religion is not". About Islam he said: "I like Islam, it is a consistent [or consequential] idea of religion and open-minded.

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