FOM: 81:Finite Distribution

Harvey Friedman friedman at math.ohio-state.edu
Mon Mar 13 01:44:51 EST 2000

```We prove a theorem concerning the relative distribution of finite subsets
of N and their images under multivariate functions from N into N.

N is the set of all nonnegative integers. Let f:N^k into N. For A
containedin N, write fA for the set of all values of f at arguments from A.
We write |A| for the cardinality of A.

We begin with the infinite form.

THEOREM 1. For all k >= 1 and f:N^k into N there exists infinite A
containedin N such that for all r >= 0,

|fA intersect [0,r]| <= (|A intersect [0,r]|+k)^k <= r.

Here is the semifinite form.

THEOREM 2. For all k,p >= 1 and f:N^k into N there exists  A containedin N,
|A| = p, such that for all r >= 0,

|fA intersect [0,r]| <= (|A intersect [0,r]|+k)^k <= r. Furthermore, there
is an upper bound on the elements of A that depends on k,p and not on f.

Here is the finite form.

THEOREM 3. For all k,p >= 1 there exists t >= 1 such that the following
holds. Let f:[0,t]^k into [0,t]. There exists A containedin [0,t], |A| = p,
such that for all 0 <= r <= t,

|fA intersect [0,r]| <= (|A intersect [0,r]|+k)^k <= r.

THEOREM 4. Theorem 3 is not provable in Peano Arithmetic (PA). In fact,
Theorem 3 implies the 1-consistency of each specific finite fragment of PA,
over EFA (exponential function arithmetic). The growth rate of the least t
as a function of k,p is just beyond the provably recursive functions of PA.
Theorem 2 (both forms) is not provable in ACA_0. In fact, Theorem 2 (both
forms) implies the 1-consistency of each specific finite fragment of PA,
over RCA_0. The growth rate of the least upper bounds in Theorem 2 as a
function of k,p is just beyond the provably recursive functions of PA.
Furthermore, these results hold if in Theorem 3, we allow the codomain of f
to be N.

Many years ago, we proved that very little is needed about the distribution
of A,fA in the case where f is a mapping from the k element subsets of N
into N. This is worth mentioning here in this context.

Let S_k(A) be the set of all k element subsets of A. For f:S_k(N) into N
and A containedin N, define fA to be the set of all values of f at k
element subsets of A.

THEOREM 5. For all k,p >= 1 and f:S_k(N) into N there exists infinite A
containedin [p,infinity) such that

|fA intersect [0,min(A)]| <= 1.

THEOREM 6. For all k,p,r >= 1 and f:S_k(N) into N there exists finite A
containedin [p,infinity), |A| = r, such that

|fA intersect [0,min(A)]| <= 1. Furthermore, there is an upper bound on the
elements of A that depends on k,p,r and not on f.

THEOREM 7. For all k,p,r >= 1 there exists t >= 1 such that the following
holds. Let f:[0,t]^k into [0,t]. There exists A containedin [p,t], |A| = r,
such that

|fA intersect [0,min(A)]| <= 1.

THEOREM 8. Theorem 7 is not provable in Peano Arithmetic (PA). In fact,
Theorem 7 is provably equivalent to the 1-consistency of PA over EFA
(exponential function arithmetic). The growth rate of the least t as a
function of k,p,r is just beyond the provably recursive functions of PA.
Theorem 6 (both forms) is not provable in ACA_0. In fact, Theorem 6 (both
forms) is provably equivalent to the 1-consistency of PA over RCA_0. The
growth rate of the least upper bounds in Theorem 6 as a function of k,p,r
is just beyond the provably recursive functions of PA. Theorem 5 is
provably equivalent to the infinite Ramsey theorem over RCA_0. Furthermore,
these results hold if in Theorem 7, we allow the codomain of f to be N.

**********

This is the 81st 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
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
50:Enormous Integers/Number Theory  7/17/99  11:39PN
51:Enormous Integers/Plane Geometry  7/18/99  3:16PM
52:Cardinals and Cones  7/18/99  3:33PM
53:Free Sets/Reverse Math  7/19/99  2:11PM
54:Recursion Theory/Dynamics 7/22/99 9:28PM
55:Term Rewriting/Proof Theory 8/27/99 3:00PM
56:Consistency of Algebra/Geometry  8/27/99  3:01PM
57:Fixpoints/Summation/Large Cardinals  9/10/99  3:47AM
57':Restatement  9/11/99  7:06AM
58:Program A/Conjectures  9/12/99  1:03AM
59:Restricted summation:Pi-0-1 sentences  9/17/99  10:41AM
60:Program A/Results  9/17/99  1:32PM
61:Finitist proofs of conservation  9/29/99  11:52AM
62:Approximate fixed points revisited  10/11/99  1:35AM
63:Disjoint Covers/Large Cardinals  10/11/99  1:36AM
64:Finite Posets/Large Cardinals  10/11/99  1:37AM
65:Simplicity of Axioms/Conjectures  10/19/99  9:54AM
66:PA/an approach  10/21/99  8:02PM
67:Nested Min Recursion/Large Cardinals  10/25/99  8:00AM
69:Baby Real Analysis  11/1/99  6:59AM
70:Efficient Formulas and Schemes  11/1/99  1:46PM
71:Ackerman/Algebraic Geometry/1  12/10/99  1:52PM
72:New finite forms/large cardinals  12/12/99  6:11AM
73:Hilbert's program wide open?  12/20/99  8:28PM
74:Reverse arithmetic beginnings  12/22/99  8:33AM
75:Finite Reverse Mathematics  12/28/99  1:21PM
76: Finite set theories  12/28/99  1:28PM
77:Missing axiom/atonement  1/4/00  3:51PM
79.Axioms for geometry  1/10/00  12:08PM
80.Boolean Relation Theory  3/10/00  9:41AM

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