[FOM] RE:2^1000

[Matt Insall]

Vladimir Sazonov wrote:


<<Note, that the expression 2^1000 also denoting this number is even
shorter!

The problem is that I should tell explicitly that I, quite naturally,
assume natural numbers in unary notation (3=|||, etc.; please, modify
and rerun your program even for 2^100; what means "the number of
electrons" or the like?), and probably something more...
See also my reply in FOM to Arnon Avron.>>





Suppose I say that I prefer the following notation:
| means 1,
| || means 2,
| || ||| means 3,
| || ||| |||| means 4,
etc.

Then what is infinity for me?  Note that in this notation system, the
``number'' of hashmarks required to describe the ``number'' N>0 is

1+(1+2)+(1+2+3)+(1+2+3+4)+...+(1+2+3+...+N),

which is (usually) larger than N by a longshot.  Not all ``abbreviation
systems'' provide representations for reality that are more feasible than
the ``unabbreviated'' system.  However, note that this very inefficient
system provides a high level of ``distinguishability'' between ``numbers'',
so that two large numbers written near to one another are easy to (visually)
compare in size.  This is a feature that using only one hashmark (and no
blanks) to distinguish between adjacent numbers does not have.  For
instance, in the unary system with only one hashmark, it is difficult to
tell which of the following two numbers is larger (or if they are equal):

|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

vs

||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Which is preferable:  an almost unreadable representation of a ``feasible''
universe or an understandable representation of the universe (``feasible''
or not) with a number system with a high degree of ``distinguishability''
between different ``numbers''?  (This, I think, is basically a version of
the problem of Sorites, with bundles of hashmarks instead of piles of sand
particles.)  But if someone thinks that the only numbers that exist are
those that can be written down using hashmarks, then why not use the system
I describe instead of the unary one described by Sazonov?


Dr. Matt Insall
Associate Professor of Mathematics
Department of Mathematics and Statistics
University of Missouri - Rolla
Rolla MO 65409-0020

insall at umr.edu
(573)341-4901