[FOM] 584: Finite Continuation Theory 3

Fri Jun 26 19:51:43 EDT 2015

We proceed more abstractly with Finite Continuation Theory, using
finite systems.

DEFINITION 1 A k-system is an (A,<,R), where (A,<) is a finite linear
ordering and R is a subset of A^k. A full continuation of (A,<,R) is
an (A',<',R'), where
i. (A,<,R) is a subsystem of the k-system (A',<',R'), in the sense
that A containedin A', < containedin <', R containedin R'.
ii. Every ordered pair from R' is order equivalent to some ordered pair from R.
iii. Suppose every ordered pair from R' union {x} is order equivalent
to some ordered pair from R, x in A'^k. Then x is in R'.

In ii we have used order equivalence for concatenations. I.e., for
2k-tuples. We also use such concatenations in Definition 2 below.

THEOREM 1. (EFA) Every k-system has a full continuation. We can take
the domain of the full continuation to be the same as the domain of
the given k-system, or any other finite superset of the domain of the
given k-system.

We now look at symmetrically full continuations.

DEFINITION 2 Let (A,<,R) be a k-system. A symmetrically full continuation of
(A,<,R) is an (A',<',R',c_1,...,c_k), where
i. (A,<,R) is a substructure of the k-system (A',<',R'), and c_1 <'
... <' c_k = max(A').
ii. Every ordered pair from R' is order equivalent to some ordered pair from R.
iii. Suppose every ordered pair from R' union {x} is order equivalent
to some ordered pair from R, max(x) <' c_j. Then  (x,c_1,...,c_k-1) is
order equivalent to some (y,c_1,...,c_k-1) and
(y,c_1,...,c_j-1,c_j+1,...,c_k), y in R'.

There are stronger notions of symmetry that can be used here, but this
notion is relatively transparent.

PROPOSITION 1. Every k-system has a symmetrically full continuation.

Proposition 1 is explicitly Pi02. There is an a priori double
exponential upper bound on the size of the continuation relative to k
and the size of the given k-system, yielding an explicitly Pi01 form.

THEOREM 2. Proposition 1 is provably equivalent to Con(SRP) over WKL_0.

Thus the injection of a transparent form of symmetry or
indiscernibility into the trivial Theorem 1 results in a Pi01 sentence
that can and can only be proved going well beyond ZFC.

************************************************************
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 584th 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

Harvey Friedman