>>10974893A bunch of analytic philosophers thought it'd be cool if they could take all of mathematics and reduce it to a language game where you come up with the rules, write it all down, and put the thing to rest once and for all.
Gödel was a Platonist and hated this analytic philosophy way of thinking.
Anyway, a couple of these philosophers set out to accomplish this goal by writing Principia Mathematica, a text that painstakingly defined everything about the foundations of mathematics they could think of e.g. It's only 160 pages or so into the thing where you get to pic related, the proof that 1+1=2.
Part of how PM was able to work was that everything was represented symbolically. So numbers, operator symbols (+, -, *, /), and any logical statement operators (both, neither, implies, therefore, etc) all got coded into this scheme.
What Gödel did was take this further and come up with "Gödel numbering," a technique for encoding PM statements with numbers using the fact that prime factorizations are unique e.g. The prime factors for the number 107 are 3, 5, and 7, and only 107 has this prime factorization. So you could take a statement, transform it into a number, and then get back to a meaning (that meaning being the provability of the statement) after performing calculations with the number.
This allowed Gödel to show numbers meaning something like "This sentence can't be proven by system X" exist which you can prove but which can't be proven by the system they're in (making the system "incomplete") without invalidating the system in the process (making the system "inconsistent").
