Zeration

[GFR]

Thank you for your strong destructive testing of my extremely long stipulations. Unfortunately, I was very busy and I didn't have time enough to make them shorter. In particular

I hope that everybody would be patient enough to read it, without ... fainting,

At least I do it!

Do you mean ... fainting? Oh, no! Try to stand under this strange situation. I told you that you would become nervous hearing about zeration. But you insisted so much. By the way, did you mean "at least" or "at last"?

You can not supplement the mother law, it alone prescribes that a[0]b=b+1 for all b.

My problem is that "for all b" means also "for any a", which, as you have seen, disturbs me a lot. In fact, in my opinion, this would mean that zeration does not exist at all, as a binary operation. And this would be very sad, at least for me.

And no wonder (... that the Ackermann Function ... may) ... lead also here to the conclusion that a[0]b=b+1.

So dry? Only to that, for any "a"? Are you sure? Unfortunately, I am not. I always understood "b+1" as a starting point in the definition of AF and not as a conclusion. But, ... Henryk, I am not a recursivist and, therefore, I might be wrong.

y = x /[s]2 ---> y <= (y[s-1](y[s]\ x)) /[s-1]2
... should read (y[s-2](y[s-1]\ x)) /[s-1]2 right?

Right! Sorry!

So let us see why this formula actually computes the value of x/[s]2.
It is in iteration formula, so if it has a limit y then
y = (y[s-2](y[s-1]\ x)) /[s-1]2

If we further assume that all arguments are in its bijectivity domain this formula is equivalen to:
y [s-1] 2 = y [s-2] ( y [s-1]\ x )
y [s-2] (y[s-1]1) = y [s-2] (y [s-1]\ x )
y[s-1]1=y [s-1]\ x
y [s-1] (y[s-1] 1) = x

And now under the assumption that y[s]0=1:
y [s] 2 = y [s] (1+1+0) = y[s-1](y[s-1](y[s]0)) = x
y = x [s]/ 2

Which, of course, should read y = x/ [s]2. You see? In the best families!. We just invented it and we still don't have enough experience with that. But, ... ... Henryk! This is the Slash Algebra. I have to think about these very nice manipulation, sleep on that and tell you later what I think. Neverthaless, for the moment, it seemed to me a very correct ... demonstration.

However the assumption is wrong for s=2, y[2]0=0.

Yes, it is wrong, but I don't see the point. Let me read the entire stuff again and again. I shall answer to you ... later. For the moment, I just wish to remind, for our further discussions, that we indeed have:
y = b[4]1 = b .... and .... y = b[4]0 = 1
y = b[3]1 = b .... and .... y = b[3]0 = 1
y = b[2]1 = b .... and .... y = b[2]0 = 0
y = b[1]1 = b+1 . and .... y = b[1]0 = b

So, we should not be surprised reading:
y = b[0]1 = b+1 .... for b>1 ... !
y = b[0]0 = b+1 .... for b>0 ... !
y = b[0]b = b+2
y = 2[0]2 = 4 ....... for b=2 ... !!!
y = 0[0]0 = 2 ....... for b=0 ... !!!!!!

Konstantin Rubtsov (Rubcov) knows a complicated, but very "clean" demonstration of the commutativity of zeration,

However he did not cleanly state from what laws this commutativity follows. Surely not from the mother law.

Surely not. It is based on metamathematical reasoning.

Pillar 4 - The Hyper-means.

The hyper means (a[n]a)/[n+1] 2 = a
follow directly from the assumption that a[n]a=a[n+1]2 which is equivalent to a[n+1]1=a.

Stop a moment, please! I don't believe in a generalisation of: a[n+1]1 = a. In fact, for n=0, a[1]1 = a+1 = a is wrong!

Asserting that a[0]a=a+2 is a bit like asserting that a[1]1=a.
Do you see the similarity?

Not really! At level a[s]1, we have different behaviours. In fact, as we have seen:
a[2]1 = a*1 = a
a[1]1 = a+1 (and tht's it!)
while:
a[2]a = a*a = a^2
a[1]a = a+a = a*2

.........
The general law is a[s]2=a[s-1](a[s]1).
If (and only if) a[s]1=a then a = (a[s-1]a)/[s] 2.
This is not satisfied for s=1, a[1]1=a+1\( \neq \)a, and hence the conclusion a=(a[0]a)/[1] 2 is wrong.
.........
If \( \lim_{t\to\infty} \) x /[n+1] t = a then a[n]x=x[n+1]1 (or equivalently (x[n+1]1) /[n] x = a).
If (and only if) x[n+1]1=x then a[n]x=x.
x[1]1\( \neq \)x hence the conclusion that a[0]x=x is wrong.

Gianfranco, I know you and KAR invested a lot into the development of your zeration. But sometimes its just time to let go.

These are two important points, based on tyhe Mother Law, and a friendly advice, which I shall seriously take into consideration. But, Henryk, the problem is that the formula mentioned in Pillar 4 may work, and the choice of -oo as neutral element of zeration (as we defined it) is shared by other researchers in this field. Would that mean that tha Mother Law is not completely right (or ... sufficient) ? I don't dare to think of that, despite the fact that we (KAR & GFR) admitted this possible limited catastrophe as a working hypothesis! What a life!

However, I agree with you that this controversial subject is out of the scope of the present Forum and suggest to perhaps stop here our discussions. Perhaps, later. we shall have more arguments to finally destroy or .., resuscitate it. Thank you for your attention.

Gianfranco