2
2. THE ACKERMANN HIERARCHY
There is a good notation for really big numbers - up to a
point. We use a streamlined version of the Ackerman
hierarchy.
Let f:Z
+
Z
+
be strictly increasing. We define the critical
function f’:Z
+
Z
+
of f by: f’(n) = the result of applying f
n times at 1.
Define f
1
:Z
+
Z
+
to be the doubling function, and f
n+1
:Z
+
Z
+
be f
n
’.
Thus f
1
is doubling, f
2
is exponentiation, f
3
is iterated
exponentiation; i.e., f
3
(n) = E*(n) = an exponential stack of
n 2’s. f
4
is confusing.
We can equivalently present this by the recursion equations
f
1
(n) = 2n, f
k+1
(1) = f
k
(1), f
k+1
(n+1) = f
k
(f
k+1
(n)), where k,n =
1. We define A(k,n) = f
k
(n).
Note that A(k,1) = 2, A(k,2) = 4. For k
≥
3, A(k,3) > A(k-
2,k-2), and as a function of k, eventually strictly dominates
each f
n
, n
≥
1.
A(3,4) = 65,536. A(4,3) = 65,536. A(4,4) = E*(65,536). And
A(4,5) is E*(E*(65,536)).
It seems safe to assert, e.g., that A(5,5) is incomprehensib-
ly large. We propose this number as a sort of benchmark.
3. VECTOR REDUCTION
Let k
≥
1 and x N
k
. We perform the "reduction" on x =
(x
1
,...,x
k
) as follows. Find the greatest i < k such that x
i
>
0, and replace x
i
,x
i+1
by x
i
-1,x
1
+...+x
k
.
THEOREM 3.1. For all k
≥
1 and x N
k
, this reduction can be
performed only finitely many times.
The number of times this reduction can be performed at x N
k
is very large. E.g.,
THEOREM 3.2. The number of times this reduction can be
performed at (2,0,0,0,0) is greater than E*(2
1,000,000
).