2
2. THE ACKERMAN 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.
For n ≥ 1, the n-th function of the Ackerman hierarchy is the
result of applying the ’ operator n-1 times starting at the
doubling function.
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(3,5) = 2
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 incomprehen-
sibly large. We propose this number as a sort of benchmark.
3. BOLZANO WEIERSTRASS
We start with the usual statement of BW.
THEOREM 3.1. Let x[1],x[2],.
.. be an infinite sequence from
the closed unit interval [0,1]. There exists k
1
< k
2
< .
..
such that the subsequence x[k
1
],x[k
2
],.
.. converges.
We can obviously move towards estimations like this:
THEOREM 3.2. Let x[1],x[2],.
.. be an infinite sequence from
the closed unit interval [0,1]. There exists k
1
< k
2
< .
..
such that |x[k
i+1
]-x[k
i
]| < 1/i
2
, i
≥
1.
But now we shake things up: