I am interested, from a lay man's perspective mainly, in the incompleteness theorem but not sure I understand even on a general level what is being said. From what I recall, it states something like the following: in all formal mathematical systems there are certain statements that can't be proven. Firstly, I could not even find a definition of what is meant by "formal mathematical system". Secondly what does the theorem mean on a practical level? Does it mean we can't be sure of anything? Does it mean there are certain problems we will never be able to solve? If the later is true, which kind of problems.
Thank you!
Thank you!