# [FOM] 593: Finite Continuation Theory 12/perfect?

Harvey Friedman hmflogic at gmail.com
Mon Aug 3 23:47:29 EDT 2015

```The perfect implicitly Pi01 sentence from
http://www.cs.nyu.edu/pipermail/fom/2015-July/018859.html

PROPOSITION A. Every finite subset of Q^k|>n has a maximal nonnegative
continuation, where S_1...n|>n = S_0...n-1|>n.

>From http://www.cs.nyu.edu/pipermail/fom/2015-July/018859.html

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 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.

I noticed Proposition B after
http://www.cs.nyu.edu/pipermail/fom/2015-July/018859.html appeared.

Of course, as written, Proposition B is Pi11 and not Pi01.

>From http://www.cs.nyu.edu/pipermail/fom/2015-July/018859.html

PROPOSITION C. 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 C is the Proposition B of
http://www.cs.nyu.edu/pipermail/fom/2015-July/018859.html

Proposition C is explicitly Pi02. As explained in
http://www.cs.nyu.edu/pipermail/fom/2015-July/018859.html with an
obvious a priori upper bound on the size of the finite continuation
from the data, it becomes explicitly Pi01.

THEOREM. Proposition B is provably equivalent to Con(SRP) over RCA0.
Proposition C is provably equivalent to Con(SRP) over EFA.

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My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
This is the 593rd 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
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
592: Finite Continuation Theory 11/perfect?  7/29/15  4:30PM

Harvey Friedman
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