Kurt Friedrich Gödel was an Austrian, and later American, logician, mathematician, and philosopher who was born on April 28, 1906 and lived through January 14, 1978.
Ever since I took the course on Theory of Computation, this man had a tremendous impact on my mind. He was known for his mathematical work in the 20th century, which was the time when most mathematicians were concentrating on the logic and set theory concepts in mathematics. At the age of 25 he made two significant contribution to the world which turned the world of Hilbert (needs no introduction) upside down. Hilbert during that time was trying axiomatize mathematics wherein all of the truths could be proved by starting from a basic set of axioms and by mechanically applying inference rules. Gödel sounded a death knell to this effort!
Gödel first proved an interesting result where if starting from a set of axioms, if one cannot derive a contradiction then the axioms are consistent. This is known as “Gödel’s Completeness Theorem”. This means that if we carefully choose our axioms and rules any complex mathematics truth could be proved mechanically however lengthy the prove may be (10 million lines may be). This bode well with Hilbert’s effort.
Gödel a year later went to prove what are famously referred as “Gödel’s incompleteness theorems”. It states that given any consistent, computable, set of axioms, there is a true statement about the integers that cannot be proven from the axioms. Another way to say this that mathematical theories pompous enough to boast their own consistency (from with in the system) are the ones that do not have consistency.
One of his other contribution I like was about the Gödel metric/solution which is an exact solution of the Einstein field equations and it postulates the existence of closed timelike curves that would allow time travel in a universe.
Einstein and Gödel used to enjoy the company of each other and had long walks on their home after work. It is interesting to note that Gödel was first rejected US Citizenship but later was granted one after Einstein’s recommendation.
I will try and elaborate what these theorems mean and their implications in my next few articles but I wanted to remember this great man and contributions to the world at large.
To Gödel !