Talk:Pair sequence number

I have just spent quite a bit of time analyzing this system, and so far, I'm getting similar results.

My basic results:

let |A| be the ordinal for sequence A.

Theorem 1. (0,0) basically separates the sequence into summands, so for example


 * (0,0) A (0,0) B (0,0) C| = |(0,0) A| + |(0,0) B| + |(0,0) C|

Proof: The (0,0)'s serve as stopgaps; the reduction rule can never pass to the left of a (0,0). So each partition that the (0,0)'s get divided into get evaluated one after the other, so the final ordinal is just the sum.

Let [A - (b,c)] be sequence A with each term decreased by b in the first term and c in the second term. So for example [(1,1)(2,1)(3,1) - (1,1)] = (0,0)(1,0)(2,0)

Theorem 2. In a part of the sequence with no (0,0)'s, the (1,0)'s satisfy the following:


 * (0,0) A (1,0) B| = |(0,0)A| * w^|[(1,0) B - (1,0)]|

Proof: We use induction on the exponent |[(1,0)B - (1,0)]|.

Successor case: From the previous theorem, |(0,0) B| + 1 = |(0,0) B (0,0)|. So we have


 * (0,0) A| * w^|(0,0) B (0,0)| = |(0,0) A| * w^(|(0,0) B| + 1) = |(0,0) A| * w^|(0,0) B| * w = |(0,0) A (1,0) [B + (1,0)]| * w (by the induction hypothesis)

and


 * (0,0) A (1,0) [B + (1,0)] (1,0)| = sup_n |{(0,0) A (1,0) [B + (1,0)]} repeated n times| = |(0,0) A (1,0) [B + (1,0)]| * w, as desired.

Limit case: Straightforward. QED

Note that we can chain the previous theorem, so for example

= |(0,0) A| * w^(|(0,0) [B - (1,0)]| * w^|(0,0) [C - (2,0)] (1,0) [D - (2,0)]|) = |(0,0) A| * w^(|(0,0) [B - (1,0)]| * w^(|(0,0) [C - (2,0)]| * w^|(0,0)[D - (3,0)]|)) = |(0,0) A| * w^|(0,0) [B - (1,0)]| * w^w^|(0,0) [C - (2,0)]| * w^w^w^|(0,0)[D - (3,0)]|
 * (0,0) A (1,0) B (2,0) C (3,0) D| = |(0,0) A| * w^|(0,0) [B - (1,0)] (1,0) [C - (1,0)] (2,0) [D - (1,0)|

Since we can partition out (0,0)'s and (1,0)'s, we will assume no interior such pairs from now on.

Theorem 3. |(0,0) A (1,1)| is the next epsilon number after |(0,0) A|

Proof:

= sup_n (|(0,0) A| + w^|(0,0) A| + w^w^|(0,0) A| + w^w^w^|(0,0) A|...)
 * (0,0) A (1,1)| = sup_n |(0,0) A (1,0) [A + (1,0)] (2,0) [A + (2,0)] (3,0) [A + (3,0)] {n repeats}|

which will be the next epsilon number after |(0,0) A|.

Theorem 4. If B has no (1,1)'s, |(0,0) A| = epsilon_a, and |(0,0) A (1,1) B| = epsilon_(a+b), then |(0,0) A (1,1) B (2,0)| = epsilon_(a+b*w)

Proof:


 * (0,0) A (1,1) B (2,0)| = sup_n |(0,0) A (1,1) B (1,1) B (1,1) B {n repeats}| = epsilon_(a+b*w)

at this point proofs get more and more difficult, but we can notice certain patterns. Once we partition out all the (0,0)'s and (1,0)'s (as well as all the new (1,0)'s that get formed by subtraction) the remaining partitions represent epsilon numbers. (1,1) increments the epsilon argument by 1, and (2,0) multiplies the part between the last (1,1) and the (2,0) by w. So for example


 * (0,0)(1,1)(2,0)| = epsilon_w
 * (0,0)(1,1)(2,0)(2,0)| = epsilon_(w^2)
 * (0,0)(1,1)(2,0)(2,0)(2,0)| = epsilon_(w^3)
 * (0,0)(1,1)(2,0)(2,0)(2,0)(1,1)| = epsilon_(w^3 + 1)
 * (0,0)(1,1)(2,0)(2,0)(2,0)(1,1)(2,0)| epsilon_(w^3 + w)

Then (3,0) functions in the same way as (1,0) did before, representing the function w^a in the epsilon argument. So


 * (0,0)(1,1)(2,0)(3,0)| = epsilon_(w^w)
 * (0,0)(1,1)(2,0)(3,0)(3,0)| = epsilon_(w^(w^2))
 * (0,0)(1,1)(2,0)(3,0)(3,0)(2,0)| = epsilon_(w^(w^2 + 1))
 * (0,0)(1,1)(2,0)(3,0)(3,0)(2,0)(3,0)| = epsilon_(w^(w^2 + w))

so just like (0,0)(1,0)(2,0)(3,0)... goes up to epsilon_0, (0,0)(1,1)(2,0)(3,0)(4,0)(5,0)... goes up to epsilon_(epsilon_0). Thus we have


 * (0,0)(1,1)(2,0)(3,1)| = epsilon_epsilon_0
 * (0,0)(1,1)(2,0)(3,1)(4,0)(5,1)| = epsilon_epsilon_epsilon_0
 * (0,0)(1,1)(2,1)| = phi(2,0)

Just as (0,0) and (1,0) took care of addition and exponentiating by w, (1,1) and (2,0) take care of epsilon arguments. So for example


 * (0,0)(1,1)(2,1)(2,0)| = epsilon_(phi(2,0) * w)
 * (0,0)(1,1)(2,1)(2,0)(3,0)| = epsilon_(w^(phi(2,0) * w))
 * (0,0)(1,1)(2,1)(2,0)(3,0),(4,0)| = epsilon_(w^w^(phi(2,0) * w))
 * (0,0)(1,1)(2,1)(2,0)(3,1)| = epsilon_epsilon_(phi(2,0) + 1)

At this point the patterns seem to get quite complicated.


 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,0)(5,1)| = epsilon_epsilon_epsilon_(phi(2,0) + 1))
 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)| = phi(2,1)
 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(2,0)(3,1)(4,1)| = phi(2,2)
 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(3,0)| = phi(2,w)

It looks like appending (2,0)(3,1)(4,1) takes the ordinal to the next zeta value.


 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(3,0)(2,0)(3,1)(4,1)| = phi(2,w+1)
 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(3,0)(2,0)(3,1)(4,1)(3,0)| = phi(2,w2)
 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(3,0)(3,0)| = phi(2,w^2)
 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(3,0)(3,0)(3,0)| = phi(2,w^3)
 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(3,0)(4,0)| = phi(2,w^w)
 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(3,0)(4,1)| = phi(2,epsilon_0)
 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(3,0)(4,1)(5,0)(6,1)| = phi(2,epsilon_epsilon_0)
 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(3,0)(4,1)(5,1)| = phi(2, phi(2,0))


 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(3,0)(4,1)(5,1)(2,0)(3,1)(4,1)| = phi(2, phi(2,0)+1)
 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(3,0)(4,1)(5,1)(2,0)(3,1)(4,1)(3,0)(4,1)(5,1)| = phi(2, phi(2,0)*2)
 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(3,0)(4,1)(5,1)(3,0)| = phi(2, phi(2,0)*w)
 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(3,0)(4,1)(5,1)(3,0)(3,0)| = phi(2, phi(2,0)*w^w)
 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(3,0)(4,1)(5,1)(3,0)(4,0)| = phi(2, phi(2,0)*w^w^w)
 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(3,0)(4,1)(5,1)(3,0)(4,1)| = phi(2, phi(2,0)*epsilon_0)
 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(3,0)(4,1)(5,1)(3,0)(4,1)(5,0)(6,1)| = phi(2, phi(2,0)*epsilon_epsilon_0)
 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(3,0)(4,1)(5,1)(3,0)(4,1)(5,1)| = phi(2, phi(2,0)^2)
 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(3,0)(4,1)(5,1)(4,0)| = phi(2, phi(2,0)^w) = phi(2, w^w^(phi(2,0)+1))
 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(3,0)(4,1)(5,1)(4,0)(4,0)| = phi(2, w^w^(phi(2,0)+2))
 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(3,0)(4,1)(5,1)(4,0)(5,0)| = phi(2, w^w^(phi(2,0)+w))
 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(3,0)(4,1)(5,1)(4,0)(5,0)(6,0)| = phi(2, w^w^phi(2,0)+w^w))
 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(3,0)(4,1)(5,1)(4,0)(5,1)| = phi(2, w^w^(phi(2,0) + epsilon_0))
 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(3,0)(4,1)(5,1)(4,0)(5,1)(6,1)| = phi(2, w^w^(phi(2,0)*2))
 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(3,1)| = phi(2, epsilon_(phi(2,0)+1))
 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(3,1)(3,0)(4,1)(5,1)(4,1)| = phi(2, epsilon_(phi(2,0)+2))
 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(3,1)(3,1)| = phi(2, epsilon_(phi(2,0)+w))
 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(3,1)(4,0)(5,1)(6,1)| = phi(2, epsilon_(phi(2,0)*2))
 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(3,1)(4,0)(5,1)(6,1)(4,0)(5,1)(6,1)| = phi(2, epsilon_(phi(2,0)^2))
 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(3,1)(4,0)(5,1)(6,1)(5,0)(6,1)(7,1)| = phi(2, epsilon_(w^w^(phi(2,0)*2)))
 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(3,1)(4,0)(5,1)(6,1)(5,1)| = phi(2, epsilon_epsilon_(phi(2,0)+1))
 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(3,1)(4,1)| = phi(2, phi(2,1))
 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(3,1)(4,1)(3,1)(4,1)| = phi(2, phi(2,2))
 * (0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(4,0)| = phi(2, phi(2,w))

This is getting quite unwieldy. I'll think about it more later. Deedlit11 (talk) 06:16, September 29, 2015 (UTC)