[FOM] 438: Kernel Structure Theory 4

Thu Aug 19 01:32:31 EDT 2010

THIS RESEARCH WAS PARTIALLY SUPPORTED BY THE JOHN TEMPLETON FOUNDATION.

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This is a follow up to 437: Kernel Structure Theory 3 http://www.cs.nyu.edu/pipermail/fom/2010-August/014972.html

There is a typo(s) in the definition of RA in section 6 there. It  
should read
RA = {y in Q^k: there exists x in A such that R(x,y}}.
However, below. we are going to use R<A.
*********
I am still tied up with some other emergencies, but I want to  
highlight a reformulation of the Templating project that beings a  
unification of the BRT approach and Kernel Structure Theory.

Of course, Kernel Structure Theory is more concrete than BRT,  
generating equivalents of Con(SRP), and when we give the tower forms,  
they are in fact explicitly Pi01.

It should be pointed out that there is a general methodology whereby  
we can give a general conversion of the infinitary sentences arising  
from KST to provably equivalent Pi01 sentences.

We use three operators on the subsets A of Q^k.

1. Two specific unary expansion operators, A# and A*. These take any A  
contained in Q^k and create A contained in A#,A* contained in Q^k.
2. One specific upper image operator R<A. This takes any R contained  
in Q^2k and A contained in Q^k, and creates R<A contained in Q^k.

Here we initially set

A# = (fld(A) union {0})^k.
A* = A union the upper shift of A.
R<A = {y: there exists x in A such that max(x) < max(y) and R(x,y)}.

Note how elementary these three operators are, in the sense that they  
just involve order on Q, 0 in Q, and the +1 function on Q.

As in BRT, we consider the 2^16 statements in

TEMPLATE #*R<. For all order invariant R contained in Q^2k, there  
exists A contained in Q^k such that a given Boolean equation in  
A,R<A,A#,A* holds.

We know that the following instance of Template #*R< is provably
equivalent, in WKL_0, to Con(SRP):

PROPOSITION #*R<. For all order invariant R contained in Q^2k, there  
exists A contained in Q^k such that A* is contained in A = A#\R<A.

CONJECTURE #*R<. Every instance of Template #*R< is provable or  
refutable in SRP+. In fact, every instance of Template R<#* is  
provable in WKL_0 + Con(SRP) or refutable in RCA_0.

This is a manageable Conjecture. From here, we can attempt to template  
#, template *, and template R<. The plan, after establishing the  
Conjecture, is to first template *.

I have not had the time to come up with the constant dimension k that  
drives the unremoveable connection with large cardinals. Initial  
investigations suggest k = 4 or 5.

We close with a discussion of finite forms. The following is provably  
equivalent to Con(SRP) over EFA.

TINITE PROPOSITION #*R<. For all order invariant R contained in Q^2k,  
there exists finite A_1,...,A_r contained in Q^k such that for all 1  
<= i < j < p <= r, A_i* is contained in A_j = A_j#\R<A_p. Furthermore,  
we can bound the magnitudes of the numerators and denominators  
appearing in A_1,...,A_r by a double exponential in k,r.

The above can be subjected to various template investigations as well.  
We will not go into this here.

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

I use http://www.math.ohio-state.edu/~friedman/ for downloadable
manuscripts. This is the 438th 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
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
424: Well Behaved Reduction Functions 3  5/29/10  8:06PM
425: Well Behaved Reduction Functions 4  5/31/10  5:05PM
426: Well Behaved Reduction Functions 5  6/2/10  12:43PM
427: Finite Games and Incompleteness 1  6/10/10  4:08PM
428: Typo Correction in #427  6/11/10  12:11AM
429: Finite Games and Incompleteness 2  6/16/10  7:26PM
430: Finite Games and Incompleteness 3  6/18/10  6:14PM
431: Finite Incompleteness/Combinatorially Simplest  6/20/10  11:22PM
432: Finite Games and Incompleteness 4  6/26/10  8:39PM
433: Finite Games and Incompleteness 5  6/27/10  3:33PM
434: Digraph Kernel Structure Theory 1  7/4/10  3:17PM
435: Kernel Structure Theory 1  7/5/10  5:55PM
436: Kernel Structure Theory 2  7/9/10  8:42PM
437: Kernel Structure Theory 3  7/9/10  1:42AM

Harvey Friedman