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
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
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
68:Bad to Worse/Conjectures 10/28/99 10:00PM
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
78:Qadratic Axioms/Literature Conjectures 1/7/00 11:51AM
79.Axioms for geometry 1/10/00 12:08PM
80.Boolean Relation Theory 3/10/00 9:41AM
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