In this essay...

If you want a nice evening, I highly recommend a visual novel called Logicomix.

It covers Bertrand Russel's life in a popular way and brings a lot of understanding of his struggle.

I have a hard copy, but it's available on archive.org too! https://archive.org/details/Logicomix-Comic-EarlyLifeOfBertrandRussell/page/n7/mode/2up

Is this where I go "actually it took 83 pages to set up an extremely rigorous system and then a couple of lines to show you could use it to prove 1+1=2"?

If those couple of lines couldn't be determined to be true without the 83 page setup then it took all 83 pages to prove 1+1=2

The proof might be somewhat lengthy, but it is quite rigorous.

And shortly after that some other guy proved that he was wrong. More specifically he proved that you cannot prove that 1+1=2. More more specifically he proved that you cannot prove a system using the system.

Ehh...
So, it's more a case that the system cannot prove it's own consistency (a system cannot prove it won't lead to a contradiction). So the proof is valid within the system, but the validity of the system is what was considered suspect (i.e. we cannot prove it won't produce a contradiction from that system alone).
These days we use relative consistency proofs - that is we assume system A is consistent and model system B in it thus giving "If A is consistent, then so too must B".
As much as I hate to admit it, classical set theory has been fairly robust - though intuitionistic logic makes better philosophical sense. Fortunately both are equiconsistent (each can be used to imply the consistency of the other).
Yk thats something some religious folks gotta understand.
- What are you talking about, filthy infidel? My holy book contains the single, eternal truth! It says so right here in my holy book!
- Sure, but I can hear em now. "If you can't prove a system using the system, then this universe (i.e. this "system") can not create (i.e. "prove") itself! It implies the existance of a greater system outside this system! And that system is MY GOD!"
  Torturing language a bit of a speciality for the charlatan.
In logic class we kinda did prove most of the integer operations, but it was more like (extremely shortened and not properly written)
If 1+1=2 and 1+1+1=3 then prove that 1+2=3
2 was just a shortened representation of 1+1 so technically you were proving that 1+1 plus 1 equals 1+1+1.
Really fun stuff. It took a long while to reach division
- It took a long while to reach division
  and even longer to reach long division?
- Presumably you were starting with a fundamental axiom such as 1 + 1 = 2, which is the difficult one to prove because it's so fundamental
- None of that sounded fun....
- Lambda calculus be like
I like how it's valid to use "more specifically" as you're specifying what exactly he did, but in both cases those are more general claims rather than more specific ones.
Both "specifically" and "generally" would work.
Yeah, but how many pages did it take?
- As many as needed.
you cannot prove a system using the system.
Doesn't that only apply for sufficiently complicated systems? Very simple systems could be provably self-consistent.
- It applies to systems that are complex enough to formulate the Godel sentence, i.e. "I am unprovable". Gödel did this using basic arithmetic. So, any system containing basic arithmetic is either incomplete or inconsistent. I believe it is still an open question in what other systems you could express the Gödel sentence.
- I think it's true for any system. And I'd say mathematics or just logic are simple enough. Every system stems from unprovable core assumptions.