Laws and Orders

[GFR]

From GML directly follows:
a=a[s]<0>a=a[s+1]1

Actually you cant use it to define a zeration because for \( s=0 \) the above equation is wrong/not applicable. Yes, \( a=a+1 \) is wrong. The context here is what you call an equality. You have an equality (your GML) that means that the equation is true for all assignments of the contained variables from their corresponding domain of definition. Then we put the particular value \( s=0 \) and \( r=0 \) into the GML and gain the *equality* \( a=a[1]1 \), i.e. it must be true for all \( a \) of the domain of definition (which I assume is the real numbers). But instead it is not even true for one \( a \)! So the GML is not true for \( s=0 \).

Thank you for your comments, Henryk!

What I am trying to say is that:
a°a = a+2, for GML
a°a = a+1 for ML
a°a = a[0]<0>a, for GML and ML supposed together valid.
and that, therefore:
a[s]<0>a = a , for any s>0 is a right equality, but:
a[0]<0>a = a is a wrong equality.

So, your point of view is that, supposing always valid ML, GML (that you derived from ML) collapses for s=0 and, therefore, it cannot be taken as the definition of zeration. My point of view is that, if you take GML as initial "source law" and as descriptive statement for the definition of the hierarchy (in this case, ML would be one of its "properties"), then we have to accept the fact that ML may collapse for s=0 and, therefore that a°a = a+1 is a wrong equality, violating the initial general definition of the hierarchy.

I think that also Andydude agrees with that. He said that the choice between GML and ML as definition of the hierarchy is indeed a matter of choice (or something like that). Actually, we should have two versions of the "level zero" operation and what I call "zeration" is the version considering GML as the definition of the hierarchy.

In other words, I (& al.) started asking to ourselves:
"Do we have hyperoperation hierarchy definable by: a[s]<r>a = a[s+1](r+1) ?"
This means that I (& al.) started by tye GML. For r=1, we have:
........
a[4]a = a[5]2 ----> a#a = a$2 (hypothetical pentation)
a[3]a = a[4]2 ----> a^a = a#2 (our tetration research)
a[2]a = a[3]2 ----> a*a = a^2 (OK, known)
a[1]a = a[2]2 ----> a+a = a*2 (- ditto -)
a[0]a = a[1]2 ----> a°a = a+2 (this is what we were looking for)
a[-1]a = a[0]2 ---> .............. (very strange, I must admit).

Now, the answer to that could be: YES, we have such a hierarcchy, or: NO, we have not such a hierarchy, but we may have something else. If the answer is that we have two or more versions of it, ... I shall feel really in trouble. We should try with a half-rank operation hierarchy and see which is the stronger initial statement (GML or ML), allowing the "chain" to survive. I mean which is the law allowing that half-way hypothetical hierarchy to exist. I discussed with Ingolf Dahl (participant in this forum) about that, some time ago. By the way, how are you, Ingolf?

I know what I said concerning the Ackermann Function (AF). But, Henryk, shall we consider that pillar as the most fundamental of the hyperops hierarchy, or just a proof of its usefulness for the description of the AF elements. In this case, we shall not be surprise that ML faints for s=0.

Just a hypothesis. What do you think?

GFR