[FOM] 397: Free Reduction Theory 4

[Harvey Friedman]

In #396, we said the following:

> #394, #395 represent the state of the art regarding the infinite  
> statements, and the games. (The formulations for n-huge can be  
> stretched to somewhat more than rank into itself). The game  
> formulations for finite length already are Pi01 (using decision  
> procedures) and easily lend themselves to clear estimates.
>
> We have been thinking about better finite forms of the infinite  
> statements. At least in the case of the SRP level statements, we  
> have temporarily decided on the most straightforward way of going  
> about this, which we now present.

Upon yet more reflection, we present an explicitly Pi01 sentence with  
some advantages.

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

FREE REDUCTION THEORY 5
by
Harvey Friedman
March 9, 2010

We use Q for the set of all rational numbers. The upper shift of x in  
Q^k is obtained from x by adding 1 to all nonnegative coordinates.

We fix R contained in Q^2k, A contained in Q^k, and x,y in Q^k, until  
the statement of Theorem 1.1.

The norm #(x) of x is the sum of the magnitudes of the numerators and  
denominators of the coordinates of x when put in reduced form.

The upper shift of x is the result of adding 1 to all of its  
nonnegative coordinates.

We write x R y if and only if (x,y) in R. We say that y is a strict  
reduction of x if and only if max(y) < max(x) and x R y.

We say that y is a reduction of x if and only if y = x or y is a  
strict reduction of x.

We say that y is a controlled reduction of x if and only if y is a  
reduction of x and #(y) <= 8#(x)^4.

NOTE: We use 8#(x)^4 here as a safe but rather poor estimate. We will  
look into what is needed later.

We say that A is free if and only if no element of A is a strict  
reduction of any element of A.

We say that A is upper shift free if and only if A together with the  
upper shifts of its elements, forms a free set.

We say that u,v in Q^r are order equivalent if and only if for all 1  
<= i,j <= r, x_i < x_j iff y_i < y_j.

We say that R is order invariant if and only if R is a union of  
equivalence classes of Q^2k under order equivalence.

THEOREM 1.1. In every order invariant R contained in Q^2k, some free  
finite set contains a controlled reduction of the vectors of norm at  
most k^2, and the vectors in its cube of norm at most k!.

PROPOSITION 1.2. In every order invariant R contained in Q^2k, some  
upper shift free finite set contains a controlled reduction of the  
vectors of norm at most k^2, and the vectors in its cube of norm at  
most k!.

It is obvious that the finite set in Proposition 1.2 can be taken to  
consist of vectors of norm at most 8k!^4. Hence Proposition 1.2 is  
explicitly Pi01.

THEOREM 1.3. Proposition 1.2 is provable in SRP+ but not in any  
consequence of SRP consistent with EFA. Proposition 1.1 is provably  
equivalent to Con(SRP) over EFA.

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

I use http://www.math.ohio-state.edu/~friedman/ for downloadable
manuscripts. This is the 397th 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 1
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

Harvey Friedman