Completeness, Consistency, Decidability
Today I learned quite a lot about the people who invented and broke our, these days, commonly used mathematical system. From Hilbert and Conway, Turning and Gödel, and many more. The basic idea behind today’s mathematical system was that based on the right set of axioms, one could proof every mathematical equation and that this way, the system itself would be complete, consistent and decidable. Means it contains all possible true statements (completeness), it is free of contradiction (consistent) and that it’s all deterministic, means there is a way to fully calculate an equation (decidable). Turns out, that within a few years, mathematicians have ripped that idea apart and have proven that our mathematical system is neither complete nor decidable, and just maybe consistent.
I came across this, while hanging around on YouTube and running into the video about the topic, which explained it in a marvellous way. And while I was familiar with all basic principles mentioned in the video from university, I was definitely missing the historical background and was also able to tie some loose ends together in my mind, making the whole area of maths more consistent in my head. 30 Minutes that are very well spend.