[FOM] 575: Finite Games and Incompleteness
Harvey Friedman
hmflogic at gmail.com
Fri Jan 23 10:42:45 EST 2015
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.
************************************************************
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
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