Zeration

[bo198214]

The suggestion to remain cool is coming from the fact that the "rule" u = 1 is violated also by the old very respectable addition. So, we see that, taking this into consideration, we are not allowed to write
neither a ° 1 = a , nor a + 1 = a.
But, this should not be a reason for doubting of zeration.

No, its rather a reason for omitting this violated rule about the initial condition a[N]1=a. As was already shown if we omit this condition and rather focus on the quite venerable b[N-1](b[N]x)=b[N](x+1) mother rule of hyper operations, then it directly follows that a[0]x=x+1 (and also even a[N]x=x+1 for all N<0).

Which (for me!) makes a lot of sense.
We all know that the successor operation n->n+1 is the mother operation in the Peaono style definition of natural numbers and operations on them.

And the rule a[N]a=a[N+1]2 falls for me also into the category of initial condition.

Can you btw show me, how you derived the commutativity of zeration?

Via the analysis of homomorphisms (see KAR).

But we have already seen that commutativity does not follow from a[0]a=a+2 and a[0](a+x)=a+x+1 (x>1) (which includes a[0](a[0]a)=a+3, etc). As I gave a counter example of an operation that satisfies both equations but which is not commutative.

This is another weak point of chosing the condition a[0]a=a+2 for defining zeration: it does not uniquely determine zeration (while it does in andydude's first path b[N-1](b[N]x)=b[N](x+1)).

In conclusion, we came again to:
a ° b = a + 1 if a > b
a ° b = b + 1 if a < b
a ° b = a + 2 = b + 2 if a = b.

but it does not really follow from anything but is rather one consideration of andydude's second path.