[FOM] 588: Finite Continuation Theory 7

[Harvey Friedman]

We now have a new motivating idea. This could be a conceptual
breakthrough for Finite Continuation Theory.

Now let's keep everything in perspective. For Infinite Continuation
Theory, it is very hard to improve on this, at least for the moment,
from

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

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

(I think there were some typos in previous postings where I wrote Q|>n
instead of Q^k|>n).

But we are now doing Finite Continuation Theory.

The motivating idea is to work with finite continuations of finite
subsets of N, together with justifications as to why numbers are not
included.

DEFINITION 1. Let A containedin N. f is a k-continuation function for
A if and only if
i. f is a partial function that maps some n in N into a subset of {0,...,n-1}.
ii. Every set of at most k linear inequalities in k variables, with
coefficients from {0,...,k}, that has a solution in the union of the
f(n), has a solution in A.
iii. Clause ii is false for any f(n) union {n}.
The scope of f consists of the domain of f and all elements of values of f.

We can view f as a k-continuation of A with justifications for numbers
not being in the k-continuation. The associated continuation of A is,
as in i, the union of the f(n). Each f(n) union {n}, by itself,
justifies that n cannot be in the associated k-continuation of A

We use a very natural generation process.

DEFINITION 2. Let f be a CJ for A containedin N, and let E containedin
N. We generate numbers from f and E in the obvious way using two
operations. We can take n+m, and we can take the elements of f(n). f
is (E,k)-full if and only if every number generated from E using a
total of k applications of these two operations lies in the scope of
f.

PROPOSITION 1. Let n,m >> k. Every finite E containedin N has a finite
({1!,...,n!},k)-full k-CJ excluding m!-1.

Proposition 1 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 sits.

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

THEOREM 2. Proposition 1 is provably equivalent to Con(SMAH) over ACA.

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

************************************************************
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 588th 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

Harvey Friedman