[FOM] 582: Link+Continuation Theory 1

[Harvey Friedman]

I have just placed a major upgrade of the previous state of the art
report on Pi01 Incompleteness.

[1] https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
#87

Even before the ink has dried on [1], or shall I say, even before the
bytes have turned cold, there is emerging what looks like the next
generation of Incompleteness:

CONTINUATION THEORY

I have put a brief account of this in the last section 11 of [1].

But I regard [1] as perfect enough to stand on its own, and not be thrown away.

We have had this continuation idea earlier, but we didn't have certain
other crucial simplifying ideas to give it a context in which it is
striking. Now we seem to have all of the pieces needed at hand.

What is happening is that order invariant sets, roots (graphs,
cliques), and probably # bases as well, are going to be replaced by
continuations.

CONTINUATIONS cut out the middle man. See, we have the equivalence
relation of order equivalence (i.e., having the same order type). This
is used for independence statements via the intermediary or "middle
man" of order invariant sets. These are sets that respect the order
equivalence relation. CONTINUATIONS cut out this middle man.

ENTHUSIASTIC IDEA: All through mathematics, one can start with a -
generally but not always - finite incompleted structure. Then one can
focus on some properties of that structure, and look at its
continuations, in the sense of more inclusive structures that share
these properties. E.g., one can look at all extensions of the
structure that have no new "patterns". One then asks for a maximal
continuation, in this sense, which has some symmetries. WE SUSPECT
that in many circumstances, you can get such CONTINUATION THEOREMS
through and only through going way beyond ZFC.

We copy part of [1], section 11, right here:

DEFINITION 11.1. For finite sequences x,y, x*y is the concatenation of
x,y. S' is a nonnegative continuation of S containedin Q^k if and only
if S containedin S'containedinÍ Q^k|>=0 and (for all x,y in S')(there
exists z,w in S)(x*y and z*w are order equivalent).

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 have been able to show that Proposition 11.1 is provably equivalent
to Con(SRP) over WKL0, and provable in SRP for fixed n.

************************************************************
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 582nd 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

Harvey Friedman