FOM: 143:BRT/A Delta fA/A U. fA/major overhaul

[Harvey Friedman]

We make a major overhaul in the Propositions. We use "sufficiently large"
in order to suppress ugly details, and say later that there is an
associated triple exponential bound.

We again discovered that factorials are not fast enough, even though we
initially thought they were. This fix has been incorporated into the new
Propositions.

I have changed the notation T* (strictly dominating part of T) to T' since
I am also using * in the standard way for the set of finite tuples from a
set.

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I am still trying to run out of ideas.

This posting is self contained.

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*PRELIMINARIES*

We let N be the set of all nonnegative integers.

Let f:N^k into N be partially defined. For A containedin N we write fA for
the set of all values of f at arguments from A.

We write f' for the upper restriction of f, which is f restricted to {x:
f(x) > max(x)}.

We use Delta for symmetric difference, and U. for disjoint union. We use \
for set theoretic difference.

As background, we mention the following versions of the Complementation
theorem, provable in RCA0.

COMPLEMENTATION. Let k >= 1 and f:N^k into N. There exists A containedin N
such that A Delta f'A = N.

COMPLEMENTATION. Let k >= 1 and f:N^k into N. There exists A containedin N
such that A U. f'A = N.

In fact there is a unique solution A to the set equation, and that unique
solution A is infinite.

These Theorems are obviously equivalent. In fact the following conditions
are equivalent for A,B containedin N.

A Delta B = N
A U. B = N
A = N\B
B = N\A.

Let E containedin N. We write E* for the set of all finite sequences from
E, including the empty sequence.

Let f:N^k into N and x in N^p, where 0 <= p <= k-1. Then we write f_x:N^k-p
into N for the cross section of f at x, where f_x(y) = f(x,y). It is
natural to extend this definition to all x in N* by taking f_x(y) =
f(x1,...,xk) in case the length of y is at least k. In this case, we think
of f_x as a 0-ary function; i.e., a constant.

A k-ary piecewise linear transformation over N is a function of the form
T:N^k into N such that there is a partition of N^k into finitely many
pieces, where each piece is given by a finite set of linear inequalities
with coefficients from N, and where f agrees with an affine function with
coefficients from N on each piece.

The nonnegativity of the coefficients has a major effect on the affine
functions, but no effect on the linear inequalities. This is because
shifting terms from one side to the other in linear inequalities has the
effect of taking negatives of coefficients.

*INFINITARY PROPOSITIONS*

PROPOSITION 1. Let T be a piecewise linear transformation over N, and r be
sufficiently large. There exists A containedin N such that for all x in
{r^1!,r^2!,r^3!,...}*, min T_x A containedin (A Delta T'A)\{r-1}.

PROPOSITION 2. Let T be a piecewise linear transformation over N, and r be
sufficiently large. There exists A containedin N such that for all x in
{r^1!,r^2!,r^3!,...}*, min T_x A containedin (A U. T'A)\{r-1}.

Here "sufficiently large" means "sufficiently large relative to T".

*FINITARY PROPOSITIONS*

PROPOSITION 3. Let T be a piecewise linear transformation over N, and r,s
be sufficiently large. There exists finite A containedin N such that for
all x in {r^1!,r^2!, ...,r^s!}*, min T_x A containedin (A Delta T'A)\{r-1}.

PROPOSITION 4. Let T be a piecewise linear transformation over N, and r,s
be sufficiently large. There exists finite A containedin N such that for
all x in {r^1!,r^2!, ...,r^s!}*, min T_x A containedin (A U. T'A)\{r-1}.

THEOREM. Propositions 1-4 are each provably equivalent to the consistency
of MAH over ACA.

In Propositions 1-4, we can find a triple exponential expression alpha(|T|)
so that we can replace "r be sufficiently large" by r > alpha(|T|), "r,s be
sufficiently large" by r,s > alpha(|T|), and N by [0,alpha(|T|), where |T|
is the maximum of the arity and the coefficients used in the presentation
of T. This results in statements with the same metamathematical status as
Propositions 1-4. In the case of Propositions 3,4 this results explicitly
Pi-0-1 statements with the same metamathematical status as Propositions 3,4.

NOTE: MAH is ZFC + {there exists a k-Mahlo cardinal}_k. RCA0 is the
current base theory for reverse mathematics (recursive comprehension axiom
naught). EFA is exponential function arithmetic. For RCA0, see Stephen G.
Simpson's book on Reverse Mathematics (SOSOA). EFA is essentially the same
as ISigma0(exp): see Hajek and Pudlak, Metamathematics of First Order
Arithmetic. For MAH, see Aki Kanamori, The Higher Infinite.

*COMPARISON WITH DISJOINT UNION INCLUSION WORK*

How do these results fit in to the development of Boolean relation theory?

The results here are quite different than the earlier clear milestone
reached in Posting # 126, Sat, 9 Mar 2002 02:10:12. The results in #126
were the subject of the "beautiful" reactions I reported, with the
classification of the 81^2 pairs of clauses. There one has a "beautiful"
category of statements, no one of which is especially natural, but where
only one, up to symmetry, is independent of ZFC.

The results in this posting is an attempt to do two things at once:

a) to give explicitly finite independence results;
b) to give a single statement that conveys specific interesting or
intriguing or otherwise notable information.

Obviously neither #126 nor #143 is yet obsolete.

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I use http://www.mathpreprints.com/math/Preprint/show/ for manuscripts with
proofs. Type Harvey Friedman in the window.

This is the 143rd 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