Tuesday, March 07, 2017
The incompleteness theorem
Gödel's incompleteness theorems was one of the landmark theorems I learned to proof during my masters (another one: the central limit theorem but that is for another day).
Let me try my best to explain in plain text why this is considered a profound result. First, all mathematical fields of study are built up from things we call axioms: statements we believe are self-evidently true and need no proof. For example: every whole number has a number one larger than it. From a set of axioms we then proceed with logical arguments to prove other results. Consistency (i.e. no two statements contradict each other) of our axioms and its logical results are vital: without it the entire mathematical theory just breaks down and nothing will be of note.
For the most part we have no doubt about the consistency of the axioms we choose but as mathematicians proving this will be the holy grail. This is where Gödel's incompleteness theorems comes in: if our axioms are just moderately complex (as you would expect them to be for most interesting fields of study), our logical system will be unable to prove its own consistency.
Taken with a pessimistic viewpoint, being unable to prove consistency means that there is a chance our theory is inconsistent. If one day someone does indeed find out that it is inconsistent, all results proven before are moot and don't stand for much (the actual discovery of inconsistence will be of huge ramifications though).
Which brings me to my personal life rant: I loathe nothing more than blatant inconsistencies between what people profess and what they actually do.
