Two Germans exiled from their own country met each other at IAS (Institute of Advanced Studies, New Jersey). Kurt Godel, an Austrian logician, mathematician, and philosopher is considered to be one of the greatest logicians in history after Aristotle.
They met each other at IAS. They walk down the path which is later named as Philosopher’s Path.
A passerby could sometimes notice these two friends talking about their fatherland, talking about philosophy, logic but not politics.
Godel is known for his famous incompleteness theorem. In simple words it is something like this:
The barber is the “one who shaves all those, and those only, who do not shave.” The question is, does the barber shave himself?
Answering this question results in a contradiction. The barber cannot shave himself as he only shaves those who do not shave. As such, if he shaves he ceases to be the barber. Conversely, if the barber does not shave himself, then he fits into the group of people who would be shaved by the barber, and thus, as the barber, he must shave.
This is called Barber’s paradox or Russell’s paradox.
See Also | Grigori Perelman – The Saint of Mathematics
Godel’s first incompleteness theorem appeared in the paper “ON FORMALLY UNDECIDABLE PROPOSITIONS OF PRINCIPIA MATHEMATICA AND RELATED SYSTEMS.”
“Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F.”
It is worth mentioning that Bertrand Russell and E.N. Whitehead in “Principia Mathematica” tried to deduce all mathematics from logic without contradiction. Godel’s incompleteness theorem shattered everything. Within a consistent system, there is always an inconsistency.
A system is <<complete>> if all possible statements can each be either derived as theorem or it can be proven that it can’t be derived.
This system is <<consistent>> if only true statements can be derived (that is no two theorems contradicted each other).
Godel says that any formal system (which is derived from axioms) cannot be both complete and consistent. That means a system, say S (which can be proven/not proven) can happen as well as a system, say S (where the statement is only TRUE, no contradiction) both cannot exist.
Godel’s incompleteness theorem created some doubts in Einstein’s GTR (General theory of Relativity). Kurt Gödel constructed the first mathematical models of the universe in which travel into the past is, in theory at least, possible. Within the framework of Einstein’s general theory of relativity Gödel produced cosmological solutions to Einstein’s field equations which contain closed time-like curves, that is, curves in spacetime which, despite being closed, still represent possible paths of bodies.
After Einstein’s death, Godel became ever more withdrawn. At some point, he tipped over the edge. Fearful of being poisoned, he would have his wife, a former cabaret dancer, test his food. When she was no longer there, he succumbed to malnutrition.
Shounak Bhattacharya is a researcher as well as a teacher in the area of
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