[FOM] 575: Finite Games and Incompleteness

[Harvey Friedman]

THIS POSTING IS SELF CONTAINED

We have an arguable improvement on the order invariant
game in http://www.cs.nyu.edu/pipermail/fom/2015-January/018506.html

FIRST TWO CORRECTION. In
http://www.cs.nyu.edu/pipermail/fom/2015-January/018506.html In the
description of G(R,k,t) there, "either 0" should be "either t". Also
max(x) < max(y) should be max(y) < max(x).

Now we start again.

DEFINITION 1. N is the set of all nonnegative integers. Let x in
N^k. ush(t,x) is the result of adding t to all coordinates of x that
are >= t.

DEFINITION 2. Let R containedin N^2k and x in N^k. We say that y is an
R reduction of x if and only if max(y) < max(x) and x R y.

I describe the reduction game RED(R,k,t), where R containedin {0,...,kt}^2k.

Play consists of k alternating moves by Alice and by Bob. Alice's k
moves are elements of {0,...,kt}^k, the first of which is arbitrary.
Bob's response to Alice's x consists of the two plays x,ush(t,x), or
some two plays y,ush(t,y) such that y is an R reduction of x. Alice's
later moves consist of integers that appear in previous moves by Alice
or Bob. Alice wins iff none of Bob's plays is an R reduction of any of
Bob's plays.

THEOREM 1. Let R containedin {0,...,kt}^2k. Bob has a winning strategy
for RED(R,k,t) if we
remove the ush's from the rules.

PROPOSITION 2. Let R containedin {0,...,kt}^2k be order invariant and
t >= (8k)!. Bob has a winning strategy for RED(R,k,t) if Alice starts
with (t,...,t).

THEOREM 3. Proposition 2 fails if we replace t by t-1. Proposition 2
holds if we replace t by 2t.

PROPOSITION 4. Let R containedin {0,...,kt}^2k be order invariant and
t >= (8k)!. Bob has a winning strategy for RED(R,k,t) if Alice starts
with (t+1,...,t+1).

Obviously Propositions 2,4 are explicitly Pi01.

THEOREM 5. Propositions 2,3 are provably equivalent to Con(SRP) over RCA_0.

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My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
https://www.youtube.com/channel/UCdRdeExwKiWndBl4YOxBTEQ
This is the 575th 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 #1-527 can be found at the FOM posting
http://www.cs.nyu.edu/pipermail/fom/2014-August/018092.html

528: More Perfect Pi01  8/16/14  5:19AM
529: Yet more Perfect Pi01 8/18/14  5:50AM
530: Friendlier Perfect Pi01
531: General Theory/Perfect Pi01  8/22/14  5:16PM
532: More General Theory/Perfect Pi01  8/23/14  7:32AM
533: Progress - General Theory/Perfect Pi01 8/25/14  1:17AM
534: Perfect Explicitly Pi01  8/27/14  10:40AM
535: Updated Perfect Explicitly Pi01  8/30/14  2:39PM
536: Pi01 Progress  9/1/14 11:31AM
537: Pi01/Flat Pics/Testing  9/6/14  12:49AM
538: Progress Pi01 9/6/14  11:31PM
539: Absolute Perfect Naturalness 9/7/14  9:00PM
540: SRM/Comparability  9/8/14  12:03AM
541: Master Templates  9/9/14  12:41AM
542: Templates/LC shadow  9/10/14  12:44AM
543: New Explicitly Pi01  9/10/14  11:17PM
544: Initial Maximality/HUGE  9/12/14  8:07PM
545: Set Theoretic Consistency/SRM/SRP  9/14/14  10:06PM
546: New Pi01/solving CH  9/26/14  12:05AM
547: Conservative Growth - Triples  9/29/14  11:34PM
548: New Explicitly Pi01  10/4/14  8:45PM
549: Conservative Growth - beyond triples  10/6/14  1:31AM
550: Foundational Methodology 1/Maximality  10/17/14  5:43AM
551: Foundational Methodology 2/Maximality  10/19/14 3:06AM
552: Foundational Methodology 3/Maximality  10/21/14 9:59AM
553: Foundational Methodology 4/Maximality  10/21/14 11:57AM
554: Foundational Methodology 5/Maximality  10/26/14 3:17AM
555: Foundational Methodology 6/Maximality  10/29/14 12:32PM
556: Flat Foundations 1  10/29/14  4:07PM
557: New Pi01  10/30/14  2:05PM
558: New Pi01/more  10/31/14 10:01PM
559: Foundational Methodology 7/Maximality  11/214  10:35PM
560: New Pi01/better  11/314  7:45PM
561: New Pi01/HUGE  11/5/14  3:34PM
562: Perfectly Natural Review #1  11/19/14  7:40PM
563: Perfectly Natural Review #2  11/22/14  4:56PM
564: Perfectly Natural Review #3  11/24/14  1:19AM
565: Perfectly Natural Review #4  12/25/14  6:29PM
566: Bridge/Chess/Ultrafinitism 12/25/14  10:46AM
567: Counting Equivalence Classes  1/2/15  10:38AM
568: Counting Equivalence Classes #2  1/5/15  5:06AM
569: Finite Integer Sums and Incompleteness  1/515  8:04PM
570: Philosophy of Incompleteness 1  1/8/15 2:58AM
571: Philosophy of Incompleteness 2  1/8/15  11:30AM
572: Philosophy of Incompleteness 3  1/12/15  6:29PM
573: Philosophy of Incompleteness 4  1/17/15  1:44PM
574: Characterization Theory 1  1/17/15  1:44AM

Harvey Friedman