One can create faster and faster rising functions and larger and larger
number until one is blue in the face. The point is that no matter how
large a finite number n one defines, I can stand on the shoulder of
giants and do better by citing n+1 using simple addition.
Now if somehow one came up with a finite number n so large that I am not
allowed to say n+1 as if I was up against an overflow limitation similar
to that found in computers, then there would be no physical way for me
to invent or cite a larger number. So it seems that if we are to define
a largest finite number we must define it in conjunction with the number
b of bits that we are allowed to use to express this number. For a given
number of bits b the largest number would be n(b).
If we use the Ackerman series of functions we need 1 bit for addition, 2
bits for multiplication, 3 bits for exponentiation, 4 bits for tetration
etc... These bits are required in addition to the bits for the input
parameter(s) of the function.
What is the largest number of bits which are available to me to define
an Ackerman function or some other fast rising function? Possibly the
number of particles in the universe? I don't know if the fairy would be
satisfied or if I could personally herd all those bits. Is she expecting
me to hand in a piece of paper with the number written on it? Maybe then
the answer would be the number generated by the largest Ackerman
function that I can write with a very fine pen on this piece of paper.
George
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Received on Tue May 23 2006 - 00:58:54 PDT