N \rhd i = i \forall i

N – 1 \rhd i = N \forall i

N – 2 \rhd i = N – 1 for odd i, N for even i

and so on.

One nicety: i \rhd N = N \forall i.

Proof that N must be divisible by 2. For N >= 3:

(N-1) = (N-2) \rhd 1 = (N-2) \rhd (N \rhd 1) = ((N-2) \rhd N) \rhd ((N-2) \rhd) 1) = x \rhd (N-1), where x depends on the parity of N. If N is odd, then x = N-1 and x \rhd (N-1) = (N-1) \rhd (N-1) = N != N-1, so we have a contradiction. Therefore N must be even.

Similar proofs are possible to prove that N must be divisible by 4 if n >= 5, etc., although I don’t have a general proof that N must be an exact power of two.

There is at most one table for any size N.

]]>nitpick: that should be

Defining

gets you the fast-growing hierarchy, which doesn’t resolve into Knuth arrows quite as nicely.

Concerning Friedman’s function, I read over Friedman’s paper not too long ago, and discovered that the same argument that he used to prove could be used to give a lower bound for of roughly . So for example is already much more than the given lower bound of about .

]]>Anyway, thanks! I totally missed that I need to put in the word latex. ]]>

I’d heard about Laver tables a few times, but this induced me to finally broke down and tried to learn the basic idea, which for me often means writing a blog article.

Unfortunately my blog article does not mention the phrase ‘Laver table’: that’s the multiplication table for the shelf I described.

]]>]]>You can use Markdown or HTML in your comments. You can also use LaTeX, like this: $latex E = m c^2 $. The word ‘latex’ comes right after the first dollar sign, with a space after it.