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Deciding on Gödel

Writer's picture: Mishkat BhattacharyaMishkat Bhattacharya

I had mentioned in an earlier post that I would likely dedicate a separate piece to Gödel. Here it is. This post relies on several books and publications; a good one is Kurt Gödel: The Genius of Metamathematics by William D. Brewer. In brief:


Life: There were three main phases to Gödel's life (1906-1978).


i) Birth and early upbringing in Brno. I felt like an illiterate on discovering this information: when I visited Brno (Czechia) earlier this year, I did not know Gödel was born there (so was Milan Kundera, and I didn't know that either; I just knew about Mendel). So I did not visit his house. What a miss.


As a child Gödel indulged in ceaseless questioning, demonstrating what would be his lifelong conviction - that every fact about the universe should have a rational explanation.


ii) Higher education in Vienna. In Vienna, Gödel completed his undergraduate degree, then his PhD (under Hans Hahn) and then his Habilitation, a kind of postdoctoral work required for becoming a professor.


Gödel started as a physicist, possibly due to reading Goethe on optics in high school. At the University of Vienna, a charismatic professor by the name of Philipp Furtwängler taught him a course on number theory and inspired him to change over to mathematics. Furtwängler was paralyzed from the neck down and lectured from a wheelchair - shades of Stephen Hawking.


iii) The remainder of his life at the Institute for Advanced Studies in Princeton. At the IAS Gödel became close friends with Einstein.


Funny story: Einstein and Morgenstern (an economist mentioned in my post on von Neumann) went with Gödel for his US citizenship interview. Gödel had studied up, months in advance, on local governance and the US constitution. He believed he had found a logical flaw in the constitution which would allow the country to become a dictatorship, and tried to explain it to the judge. Nonetheless, he was not denied his citizenship.


Gödel became well known to the general public after the publication of Douglas Hofstadter's classic in 1979.


Work:


i) Metamathematics - This subject involves the foundations of mathematics.


A. Completeness theorem (1930): This was his PhD work. He showed (roughly speaking) that all logical deductions from the axioms of a system could be proved using those axioms.


B. Incompleteness theorems (1931): This was his Habilitation work.


a) His first incompleteness theorem showed (roughly speaking) that no axiomatic system can prove all truths about natural numbers using just those axioms.


b) His second incompleteness theorem showed that such a theory could not be consistent.


You can get a flavor of the terminology and notation involved in the discussions here. I am not a trained logician, and I cannot claim to have followed the proofs in detail. What I was able to follow was the trick by which Gödel made truths verifiable in arithmetic: he assigned every logical operation on natural numbers (not, plus, times, equals...) a number. Then any symbolic logical statement could be expressed as a sequence of these numbers.


Further, he used these numbers as the powers of the smallest prime numbers. So if there are 5 numbers in the logical operation, say 4,6,11,3,4 he uses the first 5 primes: 2,3,5,7,11. Then he combines and multiplies them to get 2^4 3^6 5^11 7^3 11^4.


Such numbers are today called Gödel numbers. Gödel numbers can be factorized uniquely into their prime factors - this is guaranteed by a theorem of Euclid. Gödel numbers can be large (try calculating the one above), depending on the complexity of the corresponding logical statements.


But the cool thing is we can now - after Gödelization - manipulate logical statements about numbers by using arithmetical operations on the Gödel numbers! This is a self-referential system, which can be used (roughly speaking) to verify truths. What Gödel said is that in implementing this process sometimes you'll end up with unverifiable paradoxes, by considering statements like 'this statement is false' (The link is to a useful video explanation).


ii) Computation


Gödel proposed a definition of a computable function, and investigated theorems that speeded up computation. He believed no machine could equal the human mind. I will not write more on this as computer science is far from my specialty.


iii) General relativity


Gödel proposed a rotating universe which allowed for time travel. This model consisted of an exact solution of Einstein's equations of general relativity, and was presented in a paper contributed to Einstein's Festschrift when he turned 70. There does not seem to be any experimental proof for such a universe as the one we have does not seem to be rotating; also, Gödel's universe, unlike ours, does not expand.


We will stop here since any post on Gödel must by definition be incomplete.


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