[FOM] 413: Local Basis Construction Theory 2

Tue Apr 20 01:04:20 EDT 2010

THIS RESEARCH WAS PARTIALLY SUPPORTED BY THE JOHN TEMPLETON FOUNDATION

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A conceptual breakthrough has arisen concerning the notion of well  
behavedness for functions f:E^k into N^k. Recall that in http://www.cs.nyu.edu/pipermail/fom/2010-April/014556.html 
  we used a kind of symmetry property, which is quite natural  
combinatorially. However, we can also use a property that is quite  
natural from the point of view of analysis.

There is a reasonably standard metric on N^k. For distinct x,y in N^k,  
define

d(x,y) = 2^-n

where n is the least integer that is a coordinate of exactly one of x,y.

For our purposes, although the 2^-n function is standard in such  
contexts, we can use any strictly decreasing positive function of n.  
So what is really relevant is what I call a

"comparative metric"

which linearly (but not strictly linearly) compares the "distance"  
between one pair of points with the "distance" between another pair of  
points.

I am guessing that this notion of "comparative metric" is standard,  
although it didn't turn up on searches under that name.

There is the standard notion of nonexpansive function in the presence  
of a metric. I.e., for all x,y in dom(f),

d(fx,fy) <= d(x,y).

Note that nonexpansive makes sense in any "comparative metric".  
Perhaps this is a common notion used in measurement theory, a branch  
of mathematical psychology?

BACKGROUND THEOREM. Every polynomial P:N^k into N^k, restricted to the  
sufficiently large double exponentials, or any sufficiently "thin"  
subset of N, is the union of finitely many nonexpansive functions.

So here are the independent statements formulated using the  
nonexpansive notion.

Recall the local basis constructions, which start with E, and perform  
the * operation at the first stage, where we view * as a function from  
E^k into A_1.

PROPOSITION. Every R contained in N^k x N^k has a length 3 (length k,  
length p) local basis construction with infinite E, whose first stage  
is the union of k^k nonexpansive functions.

PROPOSITION. Every R contained in N^k x N^k has a length 3 (length k,  
length p) local basis construction with r element E, whose first stage  
is the union of k^k nonexpansive functions.

PROPOSITION. For all t >> k, every R contained in {0,...,t}^k x  
{0,...,t}^k has a length p local basis construction, with r element E,  
whose first stage is the union of k^k nonexpansive functions. It is  
sufficient for t to be greater than the length 8kpr exponential stack  
of 2's.

PROPOSITION. Every order invariant R contained in N^k x N^k has a  
length p local basis construction, with r element E, whose first stage  
is the union of k^k nonexpansive functions. Furthermore, we can find E  
living below (8kp)!.

PROPOSITION. For all r > (8k)!!, every order invariant R contained in  
N^k x N^k has a length k local basis construction, starting with  
{r,r^2,...,r^k}, whose first stage is the union of k^k nonexpansive  
functions.

All of these Propositions are provable using certain large cardinals,  
but not within ZFC.

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I use http://www.math.ohio-state.edu/~friedman/ for downloadable
manuscripts. This is the 412th 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 athttp://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

Harvey Friedman