FOM: 148:Finite forms by relativization

Wed May 15 02:55:35 EDT 2002

Recall the following from posting 126.

PROPOSITION 1. For all multarivate functions f,g from N into N of expansive
linear growth, there exists infinite A,B,C containedin N such that
A U. fA containedin C U. gB
A U. gB containedin C U. gC.

And recall the following from posting 147.

PROPOSITION 2. For all strictly dominating piecewise linear functions
f,g over N, there exists infinite A,B,C containedin N such that
A U. fA containedin C U. gB
A U. fB containedin C U. gC.

The former is equivalent to the 1-consistency of MAH and the latter to the 
consistency of MAH. The climax is that among the 81 x 81 possibilities, this is 
the only choice (up to A,B,C symmetry) that throws us out of RCA0. This is true 
of both Proposition 1 and Proposition 2. 

The most straightforward way to give a finite form of Proposition 2 is to 
require that the A,B,C be concrete. The most convenient notion of concreteness 
here involves 

piecewise linear exponential functions over N.

These are in analogy with 

piecewise linear functions over N.

We first define the exponential terms over N. Every variable and every element 
of N is an exp term over N. The exponential terms over N are closed under 
addition, multiplication, and exponentiation. 

A piecewise linear exponential function over N is a multivariate function from 
N into N defined by finitely many cases, where each case is given by a finite 
set of linear inequalities over N, and which is given by an exp term over N in 
each case. 

Here is the finite form. 

PROPOSITION 3. For all strictly dominating piecewise linear functions
f,g over N, there exists infinite A,B,C containedin N such that
A U. fA containedin C U. gB
A U. fB containedin C U. gC

where A,B,C are ranges of piecewise linear exponential functions over N.

Proposition 3 is a Pi-0-3 statement equivalent to the consistency of MAH. 

We can give an explicit triple exponential bound on the presentations of A,B,C 
from the presentations of f,g. This results in a Pi-0-1 statement equivalent to 
the consistency of MAH.

When we make these relativizations, the classification of the 81 x 81 cases 
remains intact. 

SO WHAT IS NONOBSOLETE AMONG RECENT POSTINGS? 126,145,146,147,148.

*********************************************

I use http://www.mathpreprints.com/math/Preprint/show/ for manuscripts with
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This is the 147th in a series of self contained postings to FOM covering
a wide range of topics in f.o.m. Previous ones counting from #100 are:

100:Boolean Relation Theory IV corrected  3/21/01  11:29AM
101:Turing Degrees/1  4/2/01  3:32AM
102: Turing Degrees/2  4/8/01  5:20PM
103:Hilbert's Program for Consistency Proofs/1 4/11/01  11:10AM
104:Turing Degrees/3   4/12/01  3:19PM
105:Turing Degrees/4   4/26/01  7:44PM
106.Degenerative Cloning  5/4/01  10:57AM
107:Automated Proof Checking  5/25/01  4:32AM
108:Finite Boolean Relation Theory   9/18/01  12:20PM
109:Natural Nonrecursive Sets  9/26/01  4:41PM
110:Communicating Minds I  12/19/01  1:27PM
111:Communicating Minds II  12/22/01  8:28AM
112:Communicating MInds III   12/23/01  8:11PM
113:Coloring Integers  12/31/01  12:42PM
114:Borel Functions on HC  1/1/02  1:38PM
115:Aspects of Coloring Integers  1/3/02  10:02PM
116:Communicating Minds IV  1/4/02  2:02AM
117:Discrepancy Theory   1/6/02  12:53AM
118:Discrepancy Theory/2   1/20/02  1:31PM
119:Discrepancy Theory/3  1/22.02  5:27PM
120:Discrepancy Theory/4  1/26/02  1:33PM
121:Discrepancy Theory/4-revised  1/31/02  11:34AM
122:Communicating Minds IV-revised  1/31/02  2:48PM
123:Divisibility  2/2/02  10:57PM
124:Disjoint Unions  2/18/02  7:51AM
125:Disjoint Unions/First Classifications  3/1/02  6:19AM
126:Correction  3/9/02  2:10AM
127:Combinatorial conditions/BRT  3/11/02  3:34AM
128:Finite BRT/Collapsing Triples  3/11/02  3:34AM
129:Finite BRT/Improvements  3/20/02  12:48AM
130:Finite BRT/More  3/21/02  4:32AM
131:Finite BRT/More/Correction  3/21/02  5:39PM
132: Finite BRT/cleaner  3/25/02  12:08AM
133:BRT/polynomials/affine maps  3/25/02  12:08AM
134:BRT/summation/polynomials  3/26/02  7:26PM
135:BRT/A Delta fA/A U. fA  3/27/02  5:45PM
136:BRT/A Delta fA/A U. fA/nicer  3/28/02  1:47AM
137:BRT/A Delta fA/A U. fA/beautification  3/28/02  4:30PM
138:BRT/A Delta fA/A U. fA/more beautification  3/28/02  5:35PM
139:BRT/A Delta fA/A U. fA/better  3/28/02  10:07PM
140:BRT/A Delta fA/A U. fA/yet better  3/29/02  10:12PM
141:BRT/A Delta fA/A U. fA/grammatical improvement  3/29/02  10:43PM
142:BRT/A Delta fA/A U. fA/progress  3/30/02  8:47PM
143:BRT/A Delta fA/A U. fA/major overhaul  5/2/02  2:22PM
144: BRT/A Delta fA/A U. fA/finesse  4/3/02  4:29AM
145:BRT/A U. B U. TB/simplification/new chapter  4/4/02  4:01AM
146:Large large cardinals  4/18/02  4:30AM
147:Another Way  7:21AM  4/22/02