04/25/2008, 09:51 PM

andydude Wrote:I recognize that both the GML and ML as fundamental properties of hyperoperations. I also recognize that, although less intuitive, the Balanced Mother Law (BML) is very successful at producing the usual hyper1, hyper2, hyper3 as we know them. If you are responding to my comment in that thread, then I was only talking about BML, not the other two. I think so far we have established that the GML-hyper-0 is "zeration" as you define it, ML-hyper-0 is "succession" and BML-hyper-0 is the empty set.I agree. So, for the moment, I think we have the following "laws", sometime incompatible among them, particularly "near" rank "0" (I use s and r, just to see that I have correctly understood):

GML - the Grandmother Law: ...... a[s]<r>a = a[s+1](r+1)

ML.. - the Mother Law: ............. a[s](x+1) = a[s-1](a[s]x)

BML. - the Balanced Mother Law : a[s+1](2x) = (a[s+1]x) [s] (a[s+1]x)

DL ..- the Daughter Law: ........... a[s]x = x = a[s+1]oo

Within this list, I'd like to draw the general attention to a relation "naturally" derived from GML, for r=1 (by a simple mono-iteration of [s]): a[s]a = a[s+1]2, automatically valid when GML holds.

The beautiful thing about that is that, for a=2, we get: 2[s]2 = 2[s+1]2 !!!! . Now, since we know for sure that:

2[4]2 = 2[3]2 = 2[2]2 = 2[1]2 = 4, i.e.:

2#2 = 2^2 = 2*2 = 2+2 = 4

then, we can quietly (!) assume that we should always have:

NL .. - 2[s]2 = 4 , for any (... integer?) rank s. I'd like to call this the "Niece Law", because is very close to the GML.

@Ivars: In other words, we should also have (if these operations ... exist):

2[-1]2 = 4 (minusation) !! and

2[w]2 = 4 (omegation) where w (omega .. !) is the first infinite, coutable, ordinal number.

@Andydude & Jaydfox: Please add 2[w]2 = 4 to your analysis of e[w]n, with negative n, apparently giving:

e[w]n = n-1. ..... That's ... really all, Folks ... ????? ..... Much ado about nothing.

The hypothetical cases of 2[i]2 = 4, as well as of 2[0.5]2 = 2[1.5] = 2[2.5]2 = 4, should be supported by more serious considerations. However, ... why not??!! We shall see.

It would be interesting to study the feasibility of "half-way rank" hierarchy [s=(2n+1)/2, with n relative integer], with the only link to the "mathematical reality" given by the "Gauss Mean", at rank s=3/2.... . Definitely so.

GFR