Haskell code for this can be found at https://github.com/tromp/AIT/blob/master/Laver.hs

with a binary lambda calculus version at laver.lam in the same dir.

The sequence https://oeis.org/A098820 goes up to P(56) = 16. Do you have a link to Dougherty’s proof? (None of the three linked papers appeared to contain it.) Unless it depends on the large cardinal axiom, P(n) = 16 for all n for which we could possibly ever compute the full table, regardless of whether the large cardinal axiom is consistent with ZFC.

]]>If you just start working it out for arbitrary N = 2^n, you can see that it’s quite straightforwardly computable. Compute N \rhd i for i from 1 to N. Then compute N – 1 \rhd all i, then N – 2 \rhd all i, etc.

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.

]]>Cool! I’m scared to ask how the number 7198 gets into the game.

]]>Hmm, the papers you link to do not seem to say anything about when the period of a Laver table reaches 32. What facts about F(n) (the number of critical points of members of lying below ) give us lower bounds for when the periods of Laver tables reach 2^n?

]]>Heh, now I’ve started reading all your articles. :)

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 .

]]>I fixed the triangles. This is a full-service blog.

]]>Ahh! All of my triangles are backwards!

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

There are fascinating relations between large cardinals, large countable ordinals and large finite numbers. This is one of the things I’d like to understand better. Since everything we actually do involves writing finite symbol strings, it’s perhaps not so surprising, but still, getting a really clear grip on it seems hard.

]]>Yes, it’s cool! Joseph van Name’s comment on Scott’s post led me to write this post here. In his comment, he writes a short program using the Laver table idea that takes an insanely long time to halt.

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.

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