[FOM] 589: Finite Continuation Theory 8

[Harvey Friedman]

The development in

[1] http://www.cs.nyu.edu/pipermail/fom/2015-July/018804.html

is only approximately right, and can be fixed reasonably simply.
However, we have another way of developing Finite Continuation Theory
which we now greatly prefer.

Thus we continue to chip away at unwanted features of the explicitly
Pi01 statements.

We now k-continue finite sequences of nonnegative integers by finite
sequences of nonnegative integers. All sequences are indexed from 0.

DEFINITION 1. Let x,y be finite sequences from N. x[i] is the i-term
of x, indexed from 0. x|<=s is the subsequence of x consisting of its
terms <= s. y is a k-continuation of x if and only if
i. y is a finite sequence from N extending x.
ii. Every set of at most k linear inequalities in k variables, with
coefficients from {-k,,...,k}, with a solution in the terms of x,y,
has a solution in the terms of x.
y properly k-continues x if and only if (x,y) is a k-continuation of x
with a term that is not a term of x.

We now introduce an intriguing form of maximality for finite sequences.

DEFINITION 2. Let x in N^p. x is r-maximal if and only if no x[i]+x[j]
> r,  i,j in [0,sqrt(p)/k], properly k-continues x|<=x[i]+x[j].

THEOREM 1. Every x in {0,...,r}^m has a k-continuation of any length
>= m whose initial segments are r-maximal.

PROPOSITION 2. Let n,m >> k,r,t. Every x in {0,...,r,n!,...,m!}^m has
a k-continuation of length m^t whose initial segments are r-maximal.

Proposition 2 is explicitly Pi04, and easily put in explicitly Pi03
form. It is harder to put it in explicitly Pi01 form, but this can be
done replacing >> with a double exponential expression, and another
double exponential expression for where the continuation lies.

(In #587, I wrote Pi02 instead of Pi01, in the corresponding paragraph there).

THEOREM 3. Proposition 1 is provably equivalent to Con(SMAH) over ACA.

Here SMAH is ZFC with strongly Mathlo cardinals of finite order.

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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 589th 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

Harvey Friedman