[FOM] 424: Well Behaved Reduction Functions 3

Harvey Friedman friedman at math.ohio-state.edu
Sat May 29 20:06:18 EDT 2010

```THIS RESEARCH WAS PARTIALLY SUPPORTED BY THE JOHN TEMPLETON FOUNDATION.

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There has been a major simplification in the weak kind of WELL
BEHAVEDNESS needed for the independent Pi01 sentences.

I don't know why I didn't think of this, as I have used this kind of
thing in other contexts. But this is typical of this kind of research.

Here is the key idea. Let f:A into A, A contained in N^k. The relevant
weak notion of WELL BEHAVED, is that

each regressive k-composite (or p-composite) agrees at two common
distinct (x_1,...,x_k), (x_2,...,x_k+1).

NOTE: Agreeing at two places means that they are defined at the two
places and have the same value.

Rather than repeat #423: Well Behaved Reduction Functions 2 http://www.cs.nyu.edu/pipermail/fom/2010-May/014765.html

we merely list all of the Propositions in altered form. Note that with
this simplification, there is no point in considering any Propositions
with infinite conclusion.

FINITE RAMSEY THEOREM. If n >> k,m,r, then every f:[n]^k into [m]
agrees on two distinct (x_1,...,x_k), (x_2,...,x_k+1).

I had earlier analyzed the above and showed that you get the same
rough lower bounds as for the usual Finite Ramsey Theorem, and called

I also used this shift for f:[n]^k into [m] obtaining state of the art
PA level independence. See http://www.cs.nyu.edu/pipermail/fom/2008-January/012582.html

PROPOSITION 4.1. Every R contained in N^k x N^k has a finite reduction
function whose regressive k-composites (p-composites) each agree at
two common distinct (x_1,...,x_k), (x_2,...,x_k+1).

PROPOSITION 4.2. Every R contained in N^k x N^k has a reduction
function <= 2^[8k] whose regressive k-composites each agree at two
common distinct (x_1,...,x_k), (x_2,...,x_k+1).

PROPOSITION 4.3. Every R contained in N^k x N^k has a reduction
function <= 2^[8kp] whose regressive p-composites each agree at two
common distinct (x_1,...,x_k), (x_2,...,x_k+1).

PROPOSITION 4.4. Every R contained in [2^[8k]]^k x [2^[8k]]^k has a
reduction function <= 2^[8k] whose regressive k-composites each agree
at two common distinct (x_1,...,x_k), (x_2,...,x_k+1).

PROPOSITION 4.5. Every R contained in [2^[8k)]]^k x [2^[8k]]^k has a
reduction function <= 2^[8kp] whose regressive p-composites each agree
at two common distinct (x_1,...,x_k), (x_2,...,x_k+1).

PROPOSITION 5.1. Every order invariant R contained in N^k x N^k has a
finite reduction function whose regressive k-composites (p-composites)
each agree at two common distinct (x_1,...,x_k), (x_2,...,x_k+1).

PROPOSITION 5.2. Every order invariant R contained in N^k x N^k has a
finite reduction function whose regressive k-composites (p-composites)
each agree at two common distinct (x_1,...,x_k), (x_2,...,x_k+1).

PROPOSITION 5.3. Every order invariant R contained in N^k x N^k has a
reduction function <= (8k)! whose regressive k-composites each agree
at two common distinct (x_1,...,x_k), (x_2,...,x_k+1).

PROPOSITION 5.4. Every order invariant R contained in N^k x N^k has a
reduction function <= (8k)! whose regressive k-composites each agree
at some (r,r^2,...,r^k), (r^2,...,r^k+1).

PROPOSITION 5.5. Every order invariant R contained in N^k x N^k has a
reduction function <= (8k)!! whose regressive k-composites each agree
at (7k)!!,(7k)!!^2,...,(7k)!!^k), (7k)!!^2,(7k)!!^3,...,(7k)!!^k+1).

THEOREM 6.2. For all k >= 1, every order invariant R contained in N^k
x N^k has an exponential Presburger reduction function whose
regressive k-composites each agree at two common distinct
(x_1,...,x_k), (x_2,...,x_k+1).

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

manuscripts. This is the 424th in a series of self contained numbered
postings to FOM covering a wide range of topics in f.o.m. The list of
previous numbered postings #1-349 can be found at http://www.cs.nyu.edu/pipermail/fom/2009-August/014004.html
in the FOM archives.

350: one dimensional set series  7/23/09  12:11AM
351: Mapping Theorems/Mahlo/Subtle  8/6/09  10:59PM
352: Mapping Theorems/simpler  8/7/09  10:06PM
353: Function Generation 1  8/9/09  12:09PM
354: Mahlo Cardinals in HIGH SCHOOL 1  8/9/09  6:37PM
355: Mahlo Cardinals in HIGH SCHOOL 2  8/10/09  6:18PM
356: Simplified HIGH SCHOOL and Mapping Theorem  8/14/09  9:31AM
357: HIGH SCHOOL Games/Update  8/20/09  10:42AM
358: clearer statements of HIGH SCHOOL Games  8/23/09  2:42AM
359: finite two person HIGH SCHOOL games  8/24/09  1:28PM
360: Finite Linear/Limited Memory Games  8/31/09  5:43PM
361: Finite Promise Games  9/2/09  7:04AM
362: Simplest Order Invariant Game  9/7/09  11:08AM
363: Greedy Function Games/Largest Cardinals 1
364: Anticipation Function Games/Largest Cardinals/Simplified 9/7/09
11:18AM
365: Free Reductions and Large Cardinals 1  9/24/09  1:06PM
366: Free Reductions and Large Cardinals/polished  9/28/09 2:19PM
367: Upper Shift Fixed Points and Large Cardinals  10/4/09 2:44PM
368: Upper Shift Fixed Point and Large Cardinals/correction 10/6/09
8:15PM
369. Fixed Points and Large Cardinals/restatement  10/29/09 2:23PM
370: Upper Shift Fixed Points, Sequences, Games, and Large Cardinals
11/19/09  12:14PM
371: Vector Reduction and Large Cardinals  11/21/09  1:34AM
372: Maximal Lower Chains, Vector Reduction, and Large Cardinals
11/26/09  5:05AM
373: Upper Shifts, Greedy Chains, Vector Reduction, and Large
Cardinals  12/7/09  9:17AM
374: Upper Shift Greedy Chain Games  12/12/09  5:56AM
375: Upper Shift Clique Games and Large Cardinals 1graham
376: The Upper Shift Greedy Clique Theorem, and Large Cardinals
12/24/09  2:23PM
377: The Polynomial Shift Theorem  12/25/09  2:39PM
378: Upper Shift Clique Sequences and Large Cardinals  12/25/09 2:41PM
379: Greedy Sets and Huge Cardinals 1
380: More Polynomial Shift Theorems  12/28/09  7:06AM
381: Trigonometric Shift Theorem  12/29/09  11:25AM
382: Upper Shift Greedy Cliques and Large Cardinals  12/30/09 2:51AM
383: Upper Shift Greedy Clique Sequences and Large Cardinals 1
12/30/09  3:25PM
384: THe Polynomial Shift Translation Theorem/CORRECTION 12/31/09
7:51PM
385: Shifts and Extreme Greedy Clique Sequences  1/1/10  7:35PM
386: Terrifically and Extremely Long Finite Sequences  1/1/10 7:35PM
387: Better Polynomial Shift Translation/typos  1/6/10  10:41PM
388: Goedel's Second Again/definitive?  1/7/10  11:06AM
389: Finite Games, Vector Reduction, and Large Cardinals 1 2/9/10
3:32PM
390: Finite Games, Vector Reduction, and Large Cardinals 2 2/14/09
10:27PM
391: Finite Games, Vector Reduction, and Large Cardinals 3 2/21/10
5:54AM
392: Finite Games, Vector Reduction, and Large Cardinals 4 2/22/10
9:15AM
393: Finite Games, Vector Reduction, and Large Cardinals 5 2/22/10
3:50AM
394: Free Reduction Theory 1  3/2/10  7:30PM
395: Free Reduction Theory 2  3/7/10  5:41PM
396: Free Reduction Theory 3  3/7/10  11:30PM
397: Free Reduction Theory 4  3/8/10  9:05AM
398: New Free Reduction Theory 1  3/10/10  5:26AM
399: New Free Reduction Theory 2  3/12/10  9:36AM
400: New Free Reduction Theory 3  3/14/10  11:55AM
401: New Free Reduction Theory 4  3/15/10  4:12PM
402: New Free Reduction Theory 5  3/19/10  12:59PM
403: Set Equation Tower Theory 1  3/22/10  2:45PM
404: Set Equation Tower Theory 2  3/24/10  11:18PM
405: Some Countable Model Theory 1  3/24/10  11:20PM
406: Set Equation Tower Theory 3  3/25/10  6:24PM
407: Kernel Tower Theory 1  3/31/10  12:02PM
408: Kernel tower Theory 2  4/1/10  6:46PM
409: Kernel Tower Theory 3  4/5/10  4:04PM
410: Kernel Function Theory 1  4/8/10  7:39PM
411: Free Generation Theory 1  4/13/10  2:55PM
412: Local Basis Construction Theory 1  4/17/10  11:23PM
413: Local Basis Construction Theory 2  4/20/10  1:51PM
414: Integer Decomposition Theory  4/23/10  12:45PM
415: Integer Decomposition Theory 2  4/24/10  3:49PM
416: Integer Decomposition Theory 3  4/26/10  7:04PM
417: Integer Decomposition Theory 4  4/28/10  6:25PM
418: Integer Decomposition Theory 5  4/29/10  4:08PM
419: Integer Decomposition Theory 6  5/4/10   10:39PM
420: Reduction Function Theory 1  5/17/10   2:53AM
421: Reduction Function Theory 2  5/19/10   12:00PM
422: Well Behaved Reduction Functions 1  5/23/10  4:12PM
423: Well Behaved Reduction Functions 2  5/27/10  3:01PM

Harvey Friedman

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