# [FOM] 421: Reduction Function Theory 2

Harvey Friedman friedman at math.ohio-state.edu
Tue May 18 23:05:35 EDT 2010

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

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

When I looked over Reduction Function Theory 1 http://www.cs.nyu.edu/pipermail/fom/2010-May/014744.html

I suddenly realized WHY I was so excited about the SIMPLICITY of the
statements.

But what SIMPLICITY? The actual presentation is not all that simple.

The SIMPLICITY was HIDDEN, but very very very GENUINE.

We are actually looking at the following TEMPLATES (and more):

TEMPLATE. Every R contained in N^k x N^k has a reduction function
which is "well behaved" over some infinite set.

TEMPLATE. Every R contained in N^k x N^k has a finite reduction
function which is "well behaved" over some k element set.

TEMPLATE. Every order invariant R contained in N^k x N^k has a finite
eduction function which is well behaved over some k element set.

We just gave a particular ***reasonable*** notion of "well
behavedness". I.e., a particular notion of a function f:A into A being
"well behaved" over a given subset of N.

Certainly BOTH

E is k generating for f
f is lower shift invariant over E

are perfectly fine basic examples of WELL BEHAVEDNESS of f over E -
that we used in http://www.cs.nyu.edu/pipermail/fom/2010-May/014744.html

It remains only to provide a highly thematic highly motivated highly
beautiful highly interesting highly suitable notion of

f:A into A is WELL BEHAVED over E contained in N.

MOREOVER, we also want to have a notion of

ULTIMATELY WELL BEHAVED

which is demonstrably the strongest non absurd notion of well
behavedness.

And then we want to use the Propositions

TEMPLATE. Every R contained in N^k x N^k has a reduction function
which is ultimately well behaved over some infinite set.

TEMPLATE. Every R contained in N^k x N^k has a finite reduction
function which is ultimately well behaved over some k element set.

TEMPLATE. Every order invariant R contained in N^k x N^k has a finite
reduction function which is ultimately well behaved over some k
element set.

as examples of the necessary use of large cardinals in the finite.
They are going to be equivalent to Con, and the last one (and some
others) are explicitly Pi02 and become explicitly Pi01 by putting a
double exponential bound on the f and E.

Our next postings will be titled

REDUCTION FUNCTIONS/WELL BEHAVEDNESS 1-?

as we move the investigation to the

THEORY OF WELL BEHAVEDNESS.

After this is finished, we move back to measurable cardinals through
ranks into ranks.

We close with the correction of some typos in http://www.cs.nyu.edu/pipermail/fom/2010-May/014744.html

1. The title of section 4 should be INFINITE REDUCTION FUNCTION THEOREM.
2. The title of section 5 should be FINITE REDUCTION FUNCTION THEOREM.
3. The title of section 6 should be ORDER INVARIANT REDUCTION FUNCTION
THEOREM.
4. We wrote

> We say that f:A into A is lower shift invariant over E contained in N
> if and only if for all x,y in A and 1 <= i <= k, if the E shift of x
> is y and f_i(x) < min(x), then f_i(x) = f_i(y). Here f_i is the i-th
> coordinate function of f.

Although this is technically correct given the definition of E shift
of vectors made earlier, I meant to write

We say that f:A into A is lower shift invariant over E contained in N
if and only if for all x,y in E^k and 1 <= i <= k, if the E shift of x
is y and f_i(x) < min(x), then f_i(x) = f_i(y). Here f_i is the i-th
coordinate function of f.

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

manuscripts. This is the 421st 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

Harvey Friedman

```