User blog:LittlePeng9/Addition is commutative

For last few days I've been working on the proof that addition is commutative. It might sound like a simple thing to do, but I wanted to do it formally - you know, full and exact formality of every deduction step, starting from basic axioms and ending on the final conclusion that indeed \(\forall x\forall y:x+y=y+x\). This monster is almost 3 meters wide.



To put this short, what I do is first prove by induction that \(y+0=0+y\), then, as a lemma, I show inductively that \(x+S(y)=S(x)+y\), and ultimately I use it to show, by induction, that \(x+y=y+x\).

The whole proof was written in \(\LaTeX\), as you can guess, and used pretty much only the proof package, which included \infer command which is responsible for all the horizontal bars. In writing I used 4 macros, but the whole proof still took 6749 characters, including spaces. You can find the whole code on my pastebin.

What relation does this have to googology? First, the size - it is 2,7 meters wide. Second, it can be used to visualize one thing. Friedman has said that TREE(3) cannot be proven to exist in \(ACA_0+\Pi^1_2-BI\) when using less than \(^{1000}2\) characters. Of course, this is an enormous length, but when we look at how long the proof of commutativity of addition is, and this fact is really basic principle, we can be less surprised by the fact that this number actually has so long proof of existence.