FOM: 50:Enormous Integers/Number Theory

[Harvey Friedman]

This is the 49th 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
40:Enormous Integers in Algebraic Geometry  5/17/99 11:07AM
41:Strong Philosophical Indiscernibles
42:Mythical Trees  5/25/99  5:11PM
43:More Enormous Integers/AlgGeom  5/25/99  6:00PM
44:Indiscernible Primes  5/27/99  12:53 PM
45:Result #1/Program A  7/14/99  11:07AM
46:Tamism  7/14/99  11:25AM
47:Subalgebras/Reverse Math  7/14/99  11:36AM
48:Continuous Embeddings/Reverse Mathematics  7/15/99  12:24PM
49:Ulm Theory/Reverse Mathematics  7/17/99  3:21PM

This concerns the generation of big numbers in elementary number theory.

We consider the following property *(k,A) of integers k >= 1 and sets A of
positive integers:

1. No element of A divides any different element of A.
2. For all x in A greater than k, x divides the product of all y in A less
than x.

THEOREM 1. For all k >= 1, there are finitely many A such that *(k,A), all
of which are finite. However, this cannot be proved in RCA_0. It is
equivalent to "the Ackerman function exists" over RCA_0.

So we have defined a specific finite mathematical structure associated with
any given k >= 1: the family of all sets obeying property *(k).

Let's look at some small k. We write #(k) for the largest cardinality of
any A such that *(k,A). Note that by 2, a bound on the largest numbers
appearing in any of the A's and the number of A's can be given in terms of
#(k).
#(1) = 1.
#(2) = 1.
#(3) = 2.
#(4) = 2.
#(5) = 3.
#(6) = 3.
#(7) = 4.
#(8) = 4.
#(9) = 5.
#(10) = 7.
#(11) = 8.
#(12) = 8.
#(13) = 9.
#(14) >= 530.
#(22) >= t, where t is an exponential stack of s 2's, where s is an
exponential stack of 2^1032 2's.

NOTE: These calculations up through #(13)

THOEREM 2. Let k >= 14. #(k) is at least the unary Ackerman function at t,
where t is the number of primes <= k/2. #(k) is at most the unary Ackerman
function at k.