# [FOM] 431: Finite Incompleteness/Combinatorially Simplest

Harvey Friedman friedman at math.ohio-state.edu
Sun Jun 20 23:22:31 EDT 2010

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

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THIS POSTING IS ENTIRELY SELF CONTAINED.

Our current plans for Concrete Incompleteness aim to get yet closer to
finite game theory.

However, here we revisit an earlier approach which is aimed at the
combinatorially simplest examples of Concrete Incompleteness. See the
previous postings whose titles contain the phrase "upper shift".

Recall that we moved on from the "upper shift" approach some time ago
because we wanted to connect up with existing mathematical subjects.

Some formulations below of the upper shift approach, are simpler than
previous formulations.

Once again, there are connections with kernels in digraphs and with
finite game theory. However, that is not of concern here.

Here we just aim for the simplest formulations of independent
statements, and postpone remarks, elaborations, alternative
formulations, fixing and varying numerical parameters, and
developments into a general theory. But we couldn't resist adding
section 4.

THIS POSTING IS SELF CONTAINED.

1. Reductions and Freeness.
2. Order Invariance and Upper Shift.
3. Vector Reduction Sequences.
4. Variants.

1. REDUCTIONS AND FREENESS.

We use Q be the set of all rationals. Let R contained in Q^k x Q^k =
Q^2k. We say that y is a strict R reduction of x if and only if max(x)
> max(y) and x R y. We say that y is an R reduction of x if and only
if y  = x or y is a strict R reduction of x.

Let A,B be subsets of Q^k. We say that B is an R reduction of A if and
only if every x in the least V^k containing A has an R reduction in B.

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

2. ORDER INVARIANCE AND UPPER SHIFT.

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

We say that A contained in Q^k is order invariant if and only if for
all order equivalent x,y in N^k, x in A iff y in A.

The upper shift of A contained in Q^k is obtained by adding 1 to each
nonnegative coordinate of each element of A. E.g., the upper shift of
{(-1,1,0),(0,-2,4)} is {(-1,2,1),(1,-2,5)}.

3. VECTOR REDUCTION THEOREMS.

PROPOSITION 3.1. For every order invariant R contained in Q^2k,
{0,...,k}^k has three (or k) successive finite R reductions, which,
together with their upper shifts, have an R free union.

The norm of E contained in Q^k is the maximum of the magnitudes of the
denominators and numerators in the reduced form of the coordinates of
the elements of E.

PROPOSITION 3.2. For every order invariant R contained in Q^2k,
{0,...,k}^k has three (or k) successive R reductions of norm at most
(8k)!!, which, together with their upper shifts, have an R free union.

Here (8k)!! is a safe and convenient upper bound, which can be sharply
reduced.

THEOREM 3.3. Proposition 3.1 3.2 (both with k), without "together with
their upper shifts" is provable in EFA.

THEOREM 3.4. Propositions 3.1, 3.2 (both forms) are provable in SRP+
but not from any consequence of SRP that is consistent with EFA.
Propositions 3.1, 3.2 (both forms) are provably equivalent, over EFA,
to Con(SRP).

Here SRP+ = ZFC + "for all k there exists a limit ordinal with the k-
SRP. SRP = ZFC + {there exists a limit ordinal with the k-SRP}_k. The
k-SRP asserts that every partition of the unordered k-tuples from
lambda into two pieces has a homogeneous set that is a stationary
subset of lambda.

EFA is exponential function arithmetic.

4. VARIANTS.

In Propositions 3.1, 3.2, we can start with any finite subset of Q^k

PROPOSITION 4.1. For every order invariant R contained in Q^2k, every
finite subset of Q^k has three (or k) successive finite R reductions,
which, together with their upper shifts, have an R free union.

PROPOSITION 4.2. For every order invariant R contained in Q^2k, every
finite subset of Q^k {0,...,k}^k of norm r has three (or k) successive
R reductions of norm at most (8kr)!!, which, together with their upper
shifts, have an R free union.

THEOREM 4.3. Propositions 4.1, 4.2 (both forms) are provable in SRP+
but not from any consequence of SRP that is consistent with EFA.
Propositions 4.1, 4.2 (both forms) are provably equivalent, over EFA,
to Con(SRP).

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

manuscripts. This is the 430th 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

Harvey Friedman
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