a tad more on OR-numbrals

[Marc LeBrun]

A few tardy follow-on comments:

Many thanks to Rich Schroeppel for his clarifying explanation of the EIS 
hit.  I hope his "gapology" can be used to understand the full OR-numbral 
divisor sequence

   0 1 1 2 1 3 2 3 1 3 1 5 1 5 4 4 1 3 1 5 2 3 1 7 1 3 3 8 1 9 7 5 1 3 1 5 
1 3 1 7
   1 5 1 5 3 3 3 9 1 3 3 5 1 7 3 11 1 3 3 14 3 15 13 6 1 3 1 5 1 3 1 7 2 3 
1 5 1 3
   1 9 1 3 1 8 4 3 1 7 1 7 3 5 1 7 5 11 1 3 3 5 1 7 3 7 1 3 1 11 3 7 4 14 1 
3 3 5
   1 7 5 19 1 7 4...

which, along with a zillion others isn't in EIS yet because I can't fit the 
explanation in the "margin" (to address Antti Karttunen's complaint re. 
John Layman's interesting findings).  Maybe a numbral reference web page 
someday...

In the meantime, the two interpretations of OR-numbrals I alluded to are:

1. Sets of integers, with a 1 in the Nth bit denoting N's membership in the 
set.  Then numbral addition corresponds to set union, and multiplication 
means forming the set of all pair-wise sums.  (Maybe this would be useful 
for studying addition spectra or something).

2. Also, a homogeneous binary basis in powers of B, whose carries shift L 
places left, can be analyzed by solving
   B^N + B^N  =  B^(N+L)
to find the base B = 2^(1/L).  Thus the usual L=+1 gives vanilla binary 
B=2, L=-1 gives bit-reversed binary B=1/2, L=+2 gives the "tinker-toy" base 
B=sqrt(2), and so on.

When the carries don't shift at all addition degenerates into bitwise 
OR.  Here L=0, and we get B=2^(1/0).  So OR-numbrals are also a kind of 
"infinite" base.

But just what kind of "infinities" are these B^N=2^(N/0)?  Particularly 
that lsb "finity", 2^(0/0)?!

Can you unify these two interpretations?  A class of all sets of N things 
is somehow the same as some kind of Nth transfinite numeral?

(I suppose I should also mention that when you throw the carries away 
altogether you get XOR.  You might think of this as L=(minus) infinity, 
otherwise known as "polynomials over Z2" etc.  So perhaps either XOR is 
some kind of infinitesimal arithmetic, or else maybe a shaggy dog story 
about different flavors of zero...).

Each numbral system has its own "numbral theory" with analogs of 
partitions, divisors, etc that remain to be explored and sequenced.  If you 
come up with more such systems, interpretations and/or mysterious hits, 
please let me know.

Happy New Year!