FOM: 141:BRT/A Delta fA/A U. fA/gramatical improvement
Harvey Friedman
friedman at math.ohio-state.edu
Fri Mar 29 22:34:23 EST 2002
Propositions 5 and 6 have been gramatically changed for a better crescendo
effect. This sort of change may seem trivial, but could prove valuable.
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I am still trying to run out of ideas.
This posting is self contained.
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*TEMPLATES*
We let N be the set of all nonnegative integers and Z be the set of all
integers. For x in Z^k, we write |x| for the sup norm of x, which is the
maximum of the magnitudes of the coordinates of x.
Let f:Z^k into Z. We say that f is strictly dominating if and only if for
all x in Z^k, |f(x)| > |x|. We make the same definition with Z replaced by
N.
Let A containedin Z. We let fA be the set of all values of f at arguments
from A. We make the same definition with Z replaced by N.
Delta is symmetric difference, and U. is disjoint union.
We start with the following versions of the Complementation theorem,
provable in RCA0.
COMPLEMENTATION. Let k >= 1 and f:N^k into N be strictly dominating There
exists A containedin N such that A Delta fA = N.
COMPLEMENTATION . Let k >= 1 and f:N^k into N be strictly dominating. There
exists A containedin N such that A U. fA = 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.
We now have the following templates.
TEMPLATE. Let k >= 1 and f:N^k into N be strictly dominating and suitable.
There exists A containedin N such that A Delta fA is "big".
TEMPLATE. Let k >= 1 and f:N^k into N be strictly dominating and suitable.
There exists A containedin N such that A U. fA is "big".
But we have already made A Delta fA and A U. fA as "big" as possible. What
we really want is this.
TEMPLATE. Let k >= 1 and f:N^k into N be strictly dominating and suitable.
There exists A containedin N such that A Delta fA is "big" but "not too big".
TEMPLATE. Let k >= 1 and f:N^k into N be strictly dominating and suitable.
There exists A containedin N such that A U. fA is "big" but "not too big".
It turns out that this plan is not promising, taken literally. However,
consider the following modified templates.
TEMPLATE. Let k >= 1 and f:N^k into N be strictly dominating and suitable.
There exists A containedin N such that A Delta fA is "big relative to A"
but "not too big".
TEMPLATE. Let k >= 1 and f:N^k into N be strictly dominating and suitable.
There exists A containedin N such that A U. fA is "big relative to A" but
"not too big".
However, these templates don't quite make sense since A could be empty! So
we work with the following further modified templates.
TEMPLATE 1. Let k >= 1 and f:N^k into N be strictly dominating and suitable.
There exists nonempty A containedin N such that A Delta fA is "big relative
to A" but "not too big".
TEMPLATE 2. Let k >= 1 and f:N^k into N be strictly dominating and suitable.
There exists nonempty A containedin N such that A U. fA is "big relative to
A" but "not too big".
We endeavor to arrange that it is necessary and sufficient to use large
cardinals (or equivalent) to prove these statements.
Note that The Complementation Theorem gives us a nonempty A containedin N
such that A Delta fA and A U. fA are "big relative to A". However it is so
"big relative to A" that it is "too big", and so does not satisfy Templates
1,2.
There are two closely related Templates that we also work with.
TEMPLATE 3. Let k >= 1 and f:N^k into N be strictly dominating and suitable.
There exists "large" A containedin N such that A Delta fA is "big relative
to A".
TEMPLATE 4. Let k >= 1 and f:N^k into N be strictly dominating and suitable.
There exists "large" A containedin N such that A U. fA is "big relative to A".
The Complementation Theorem does meet Templates 3,4 with "large" being
"infinite" and "big relative to A" being "equal to N". So the plan is to
strengthen the notion of "large" and weaken the notion of "bit relative to
A".
We have apparently succeeded in carrying this out - i.e., finding natural
instances of Templates 1-4 for which it is necessary and sufficient to
use large cardinals (or equivalent).
*LINEAR NOTATION*
A k-ary piecewise linear function is a function f:Z^k into Z such that
there is a partition of Z^k into finitely many pieces, where each piece
given by a finite set of linear inequalities with integer coefficients, and
where f agrees with an affine function with integer coefficients on each
piece.
A function is piecewise linear if and only if it is k-ary piecewise linear
for some k >= 1.
Let E containedin Z. A k-ary piecewise linear function with coefficients
from E is a function f:Z^k into Z such that there is a partition of Z^k
into finitely many pieces, where
i) each piece given by a finite set of linear inequalities of the form
c1x1 + ... + ckxk + ck+1 Y d1x1 + ... + dkxk + dk+1
where c1,...,ck+1,d1,...,dk+1 in E U {0,1} and Y is among <,>,<=,>=,=,not=;
ii) on each piece, f agrees with a function of the form
c1x1 + ... + ckxk + dk+1
where c1,...,ck+1 in E U {0,1}.
*INFINITARY PROPOSITIONS*
PROPOSITION 1. Let k >= 1 and T be a strictly dominating piecewise linear
function. There exists A containedin N such that A Delta TA contains the
least value over A of every k-ary piecewise linear function with coefficients
from the infinitely many factorials lying in A.
PROPOSITION 2. Let k >= 1 and T be a strictly dominating piecewise linear
function. There exists A containedin N such that A U. TA contains the least
value on A over every k-ary piecewise linear function with coefficients from
the infinitely many factorials lying in A.
We can sharpen these as follows.
PROPOSITION 3. Let k >= 1 and T be a strictly dominating piecewise linear
function. There exists A containedin N such that A Delta TA contains the
least value over A of every k-ary piecewise linear function with coefficients
from the factorials, almost all of the latter lying in A.
PROPOSITION 4. Let k >= 1 and T be a strictly dominating piecewise linear
function. There exists A containedin N such that A U. TA contains the least
value on A over every k-ary piecewise linear function with coefficients from
the factorials, almost all of the latter lying in A.
Propositions 1-4 meet Templates 3,4.
*FINITARY PROPOSITIONS*
Here we prefer to work with Templates 1,2.
PROPOSITION 5. Let k,n >= 1 and T be a strictly dominating k-ary piecewise
linear function with unit coefficients. There exists nonempty finite A
containedin N such that the least value of any k-ary piecewise linear
function with coefficients from the first n factorials, over A, lies in A
Delta TA without (k+8)!!!-1.
PROPOSITION 6. Let k,n >= 1 and T be a strictly dominating k-ary piecewise
linear function with unit coefficients. There exists nonempty finite A
containedin N such that the least value of any k-ary piecewise linear
function with coefficients from the first n factorials, over A, lies in A
U. TA without (k+8)!!!-1.
We can still work with Templates 3,4 as follows.
PROPOSITION 7. Let k,n >= 1 and T be a strictly dominating k-ary piecewise
linear function with unit coefficients. There exists finite A containedin N
including (k+8)!!!, such that A Delta TA contains the least value over A of
every k-ary piecewise linear function with coefficients from the first n
factorials.
PROPOSITION 8. Let k,n >= 1 and T be a strictly dominating k-ary piecewise
linear function with unit coefficients. There exists finite A containedin N
including (k+8)!!!, such that A U. TA contains the least value over A of
every k-ary piecewise linear function with coefficients from the first n
factorials.
These are very conservative with the (k+8)!!! and (k+8)!!!-1 and when
things settle down, I
will see what number is really needed.
We know that the mere existence of a function of k so that any of these
Propositions holds requires large cardinals (or equivalent) to prove.
So one can fruitfully seek information about the least number, dependent
only on k, that can be used instead of (k+8)!!!-1 or (k+8)!!!. We will look
at this later.
Note that if we remove "without (k+8)!!!-1" or "including (k+8)!!!" then
the Propositions are easily provable in EFA.
By the Presburger decision procedure, or by more direct methods, we see
that strict domination of T is a finitary attribute. In addition, we can
replace N by, say, [0,(n+k+1)!!!!]. Thus we see that Propositions 5-8
become explicitly Pi-0-1.
THEOREM. Propositions 1-8 are each provably equivalent to the consistency
of MAH over ACA.
Speaking of Presburger, the results hold even if we replace "piecewise
linear" with "Presburger".
NOTE: MAH is ZFC + {there exists a k-Mahlo cardinal}_k. ACA is the system
of arithmetic comprehension with full induction. For the former see, e.g.,
Kanamori's book "The Higher Infinite". For the latter, see Simpson's book
on Reverse Mathematics (SOSOA).
*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 interseting or
intriguing or otherwise notable information.
Obviously neither #126 nor #141 is yet obsolete.
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This is the 141st 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
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