Proving the halting problem and Godel's two incompleteness theorems on one page

So I thought I'd share a proof that I first read about in Quantum Computing Since Democritus by Scott Aaronson. However, the proof can be a bit tough to follow because it's only words.  I hope this intuitive approach using diagrams will allow a greater number of individuals to understand Godel's incompleteness theorems. The proofs are short, all you really need is the concept of a computer. Two things before I begin however:

(1) I know that some of my friends who're mathematicians will be slightly grumpy about the lack of rigour. But in my opinion, rigour is the price that one has to pay for simplicity and elegance. I actually remember a quote by Nietzsche, "I wish to say within the space of ten sentences what others say in one book." Feel free to disagree though, everyones free to follow their own approach. I admit, I actually favoured more tedious and rigorous mechanisms for proving things in the past. Christopher eventually managed to convert me with his far simpler and more intuitive approaches towards problem solving. Any leanings of mine towards an elegant proof is due to an aesthetic sense that he helped develop. 

(2) Anyhow, I hope no one minds my more modern approach towards latin, quod erad demonstrandum is a bit of a mouthful, #QED should do nicely. Christopher's also the person who taught me this fine product of 21st century pop culture and I have never failed to adapt myself to the modern era since! The naming baptism of the two theorems as FIT and SIT (first incompleteness theorem and second incompleteness theorem) is simply a demonstration of my devotion towards more pop based phrases. 

(On a sidenote, just click on the pictures and zoom in to see things a bit more clearly) 

So before proving the incompleteness theorems, we must first prove the halting problem. We do that with the oldest tool in the mathematicians arsenal, contradiction. 

Fig 1: An intuitive block diagram proof for the halting problem

Alright so now that we've proved that, we shall start with Godel's first incompleteness theorem. And how do we prove that? 

Well, the answer is contradiction 

Fig 2: A proof for the first incompleteness theorem

Right, so that one requires a small bit of insight; the insight is that a statement about halting is a statement about integers. Skies are still looking blue, so we can adopt the above strategy to prove the second incompleteness theorem. Does anyone wanna guess how we'll do that? 

10 points to the person who said: "contradiction" 

Fig 3: A proof for Godel's second incompleteness theorem 

And there you have it! Personally speaking, I quite like the proof for the second incompleteness theorem. These proofs are quite elegant and they allow us to skip over the slightly gruesome way the two incompleteness theorems are taught in regular math classes (the long way). Don't worry if you don't get it right away though, spend some time thinking about it. Its not the result thats important but the manner of thinking used to achieve that result. You might have a slightly different approach to doing it but the important thing is to manifest your thoughts in some way. Reading proofs like texts won't get you very far. It's important to be able to visualise what's happening. 

I hope everyone enjoyed this!