[FOM] 592: Finite Continuation Theory 11/perfect?
Harvey Friedman
hmflogic at gmail.com
Wed Jul 29 16:30:43 EDT 2015
http://www.cs.nyu.edu/pipermail/fom/2015-July/018813.html seems to
represent the end to my ideas along certain lines. I certainly did NOT
arrive at a perfect explicitly finite statement.
I now have another approach, still within the category of Finite
Continuation Theory. It looks like a candidate for Perfection, but it
needs to sit for a while before we can make that judgement.
So let's start over from our starting point. In
[1] https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
#87
we have the "perfect" statement:
PROPOSITION 11.1. Every finite E containedin Q^k|>n has a maximal
nonnegative continuation, where S_1...n|>n = S_0...n-1|>n.
We like it slightly better to use fewer letters:
PROPOSITION A. Every finite subset of Q^k|>n has a maximal nonnegative
continuation, where S_1...n|>n = S_0...n-1|>n.
The NEW APPROACH is simply this. We preserve the above statement, but
use a weaker notion of maximality.
Here is the magic definition.
MAGIC DEFINITION. Let P be a property of subsets of X and E be an
equivalence relation on any set. We say that S is E-maximal for P if
and only if for all S' containing S, with property P, every element of
S' is E-related to an element of S.
PROPOSITION B. Let E be an order theoretic equivalence relation on Q^k
with finitely many equivalence classes. Every finite subset of Q^k|>n
has a finite E-maximal nonnegative continuation, where S_1...n|>n =
S_0...n-1|>n.
Proposition B is explicitly Pi02. There is an obvious a priori bound
on the size of S in terms of k,n and the number of E equivalence
classes. When using this bound, we obviously obtain a Pi01 sentence
using the decision procedure for (Q,<). We can also directly bound the
denominators and numerators used in S to obtain an explicitly Pi01
sentence.
Also note that if we omit "finitely many" and the last "finite", then
we get a statement obviously equivalent to Proposition A. In this way,
Proposition B is a compelling finite form of Proposition A.
THEOREM. Proposition B is provably equivalent to Con(SRP) over EFA.
************************************************************
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 591st 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
575: Finite Games and Incompleteness 1/23/15 10:42AM
576: Game Correction/Simplicity Theory 1/27/15 10:39 AM
577: New Pi01 Incompleteness 3/7/15 2:54PM
578: Provably Falsifiable Propositions 3/7/15 2:54PM
579: Impossible Counting 5/26/15 8:58PM
580: Goedel's Second Revisited 5/29/15 5:52 AM
581: Impossible Counting/more 6/2/15 5:55AM
582: Link+Continuation Theory 1 6/21/15 5:38PM
583: Continuation Theory 2 6/23/15 12:01PM
584: Finite Continuation Theory 3 6/26/15 7:51PM
585: Finite Continuation Theory 4 6/29/15 11:23PM
586: Finite Continuation Theory 5 6/20/15 1:32PM
587: Finite Continuation Theory 6 7/1/15 11:39PM
588: Finite Continuation Theory 7 7/2/15 2:44PM
589: Finite Continuation Theory 8 7/4/15 6:51PM
590: Finite Continuation Theory 9 7/6/15 5:20PM
591: Finite Continuation Theory 10 7/12/15 3:38PM
Harvey Friedman
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