FOM: 40:Enormous Integers in Algebraic Geometry

[Harvey Friedman]

This is the 40th in a series of self contained postings to fom covering a
wide range of topics in f.o.m. Previous ones are:

1:Foundational Completeness   11/3/97, 10:13AM, 10:26AM.
2:Axioms  11/6/97.
3:Simplicity  11/14/97 10:10AM.
4:Simplicity  11/14/97  4:25PM
5:Constructions  11/15/97  5:24PM
6:Undefinability/Nonstandard Models   11/16/97  12:04AM
7.Undefinability/Nonstandard Models   11/17/97  12:31AM
8.Schemes 11/17/97    12:30AM
9:Nonstandard Arithmetic 11/18/97  11:53AM
10:Pathology   12/8/97   12:37AM
11:F.O.M. & Math Logic  12/14/97 5:47AM
12:Finite trees/large cardinals  3/11/98  11:36AM
13:Min recursion/Provably recursive functions  3/20/98  4:45AM
14:New characterizations of the provable ordinals  4/8/98  2:09AM
14':Errata  4/8/98  9:48AM
15:Structural Independence results and provable ordinals  4/16/98
10:53PM
16:Logical Equations, etc.  4/17/98  1:25PM
16':Errata  4/28/98  10:28AM
17:Very Strong Borel statements  4/26/98  8:06PM
18:Binary Functions and Large Cardinals  4/30/98  12:03PM
19:Long Sequences  7/31/98  9:42AM
20:Proof Theoretic Degrees  8/2/98  9:37PM
21:Long Sequences/Update  10/13/98  3:18AM
22:Finite Trees/Impredicativity  10/20/98  10:13AM
23:Q-Systems and Proof Theoretic Ordinals  11/6/98  3:01AM
24:Predicatively Unfeasible Integers  11/10/98  10:44PM
25:Long Walks  11/16/98  7:05AM
26:Optimized functions/Large Cardinals  1/13/99  12:53PM
27:Finite Trees/Impredicativity:Sketches  1/13/99  12:54PM
28:Optimized Functions/Large Cardinals:more  1/27/99  4:37AM
28':Restatement  1/28/99  5:49AM
29:Large Cardinals/where are we? I  2/22/99  6:11AM
30:Large Cardinals/where are we? II  2/23/99  6:15AM
31:First Free Sets/Large Cardinals  2/27/99  1:43AM
32:Greedy Constructions/Large Cardinals  3/2/99  11:21PM
33:A Variant  3/4/99  1:52PM
34:Walks in N^k  3/7/99  1:43PM
35:Special AE Sentences  3/18/99  4:56AM
35':Restatement  3/21/99  2:20PM
36:Adjacent Ramsey Theory  3/23/99  1:00AM
37:Adjacent Ramsey Theory/more  5:45AM  3/25/99
38:Existential Properties of Numerical Functions  3/26/99  2:21PM
39:Large Cardinals/synthesis  4/7/99  11:43AM

A complete archiving of fom, message by message, is available at
http://www.math.psu.edu/simpson/fom/
Also, my series of self contained postings (only) is archived at
http://www.math.ohio-state.edu/foundations/manuscripts.html

FAVORITE SELF CONTAINED POSTINGS: 21, 25, 27, 31, 32, 34, 35', 37, 38, 39, 40.

This posting is prompted by a recasting of some work I had done in 1986
after digesting Simpson's result which shows the equivalence over RCA_0 of
the Hilbert basis theorem for polynomial rings in several variables over
fields with omega^omega is well ordered. See Steve Simpson, Ordinal numbers
and the Hilbert basis theorem, JSL 53 (1988), 961-974.

I gave a finite form of the Hilbert basis theorem, and gave upper and lower
bounds showing that the Ackerman hierarchy is spanned. Some years later, I
gave some further finite forms involving the degree spectrum of ideal
bases. But I now prefer the following closely related formulation in
algebraic geometry.

Let k >= 1. A k-dimensional complex algebraic set is a subset of C^k which
is the set of simultaneous zeros of a finite set of polynomials in k
complex variables with complex coefficients.

The degree of a k-dimensional complex algebraic set A is the least d such
that A can be given by complex polynomials of degree at most d.

We now make the following construction. Start with a complex algebraic set
of degree 1 in k dimensions, and successively take proper algebraic subsets
of exactly one higher degree. We continue as long as we can. By Hilbert's
basis theorem, we must stop.

QUESTION: How long can we continue this nondeterministic construction?

We let J(C,k) denote the length of the longest such construction, where k
is the dimension.

THEOREM 1. J(C,k) is finite. There are integer constants c,d > 0 such that
for all k > c, A(k) < J(C,k+c) < A(k+d), where A is the Ackerman function.

We can generalize this by starting with an algebraic set of degree n in k
dimensions, and successively take proper algebraic subsets of exactly one
higher degree. We continue as long as we can. We define J(C,k,n) as the
length of the longest such construction.

THEOREM 2. J(C,k,n) is finite. For each k, J(C,k,n), as a function of n is
primitive recursive. However, for every primitive recursive function f
there exists k such that f is strictly dominated by J(C,k,n) as a function
of n. There are integer constants c,d > 0 such that the p+c-th level of the
Ackerman hierarchy is trapped between J(C,k,p) and J(C,k,p+d).

We can further generalize these functions to arbitrary fields F. Thus
define J(F,k,n) exactly like J(C,k,n) except that we work over F instead of
C.

We established a surprising uniformity in 1986:

THEOREM 3. Fix k,n >= 1. The supremum of J(F,k,n) over all fields F is finite.

We write this supremum as J(k,n).

THEOREM 4. Theorems 1 and 2 holds for J(k,n).

It appears that the technology exists to place these functions more or less
exactly in the Ackerman hierarchy of functions, which starts with doubling,
then exponentiation, then superexponentiation, etcetera. The level is just
about the same as the dimension.

In our original formulation, we used ascending chains of ideals, where the
i-th ideal has degree at most i. And also, where the i-th ideal has degree
at most c+i. We obtained the same results.

We can also relax the condition that at the i-th stage, we have an
algebraic set of degree exactly i. This will not change any of the results.

Now for a big number.

J(C,4,4) is the longest possible length starting with a complex algebraic
set in 4 dimensions of degree 4, and successively taking proper algebraic
subsets of exactly one higher degree. The result will still hold for the
complex numbers instead of R.

THEOREM 5. J(R,4,4) is greater than 2^2^2^2^2^2 = 2^2^2^16 = 2^2^65,536.