[FOM] 597: Finite Continuation Theory 16/perfect?

Harvey Friedman hmflogic at gmail.com
Mon Aug 10 22:26:00 EDT 2015

Recall the Perfect Incompleteness:

PROPOSITION A. Every finite subset of Q^k|>n has a maximal nonnegative
continuation, where S[1,...,n]|>n = S[0,...,n-1]|>n.

PROPOSITION B. Every finite subset of Q^k|>n has a maximal nonnegative
r-continuation, where S[1,...,n]|>n = S[0,...,n-1]|>n.

PROPOSITION C. Every finite length tower of finite subsets of Q^k|>n
has a maximal nonnegative r-continuation tower, where each set
satisfies S[1,...,n]|>n = S[0,...,n-1]|>n.

We now have two main approaches to finite forms of Proposition A. The
first is Con(SRP). The first is a new version which looks very good
right now. It is Con(SMAH).

The second is Con(SRP), and is a somewhat simplified form of
http://www.cs.nyu.edu/pipermail/fom/2015-August/018871.html We will
keep it in play until I am more comfortable with the first version.

Both versions come in three variants, and use four kinds of
COMPLEMENTS of continuations. We need to define the last two of these
four complementations more carefully. It is most convenient to have
them depend on what is being continued. View these modified
definitions as corrections to

Also it is convenient to define continuations slightly differently to
handle degenerate cases properly.

DEFINITION 1. [t] = {1,2,...,t}. Let A,B containedin Z+^r, A is
without p if and only if p is not a coordinate of any element of A.
A,B are order equivalent if and only if every element of A (B) is
order equivalent to an element of B (A). A,B are order equivalent over
m_1,...,m_r if and only if A x {(m_1,...,m_r)} and B x {(m_1,...,m_r)}
are order equivalent.

DEFINITION 2. Let A containedin [r]^k. S is a continuation of A in
[t]^k if and only if A containedin S containedin [t]^k union A, where
every concatenated ordered pair from S is order equivalent to some
concatenated ordered pair from A.

DEFINITION 3. Let A containedin [r]^k and S be a continuation of A in
[t]^k. C(S) = {x in [t]^k: x not in S}. CC(S) = {x in [t]^k: S union
{x} is not a continuation of S}.

DEFINITION 4. Let A containedin [r]^k and S containedin [t]^k.
CC(S,A,<=) = {x in [t]^k: S|<=max(x) union A union {x} is not a
continuation of A in [t]^k}. CC(S,A,<) = {x in [t]^k: S|<max(x) union
A union {x} is not a continuation of A in [t]^k}.

CC is read "continuation complement".

C(S) = CC(S,<=), C(S) = CC(S,<) each indicate a stronger directional
kind of maximality of S as a continuation in its ambient space.

THEOREM. Every subset of [k]^k has a continuation in [t!]^k, where
C(S) = CC(S). Same with CC(S,A,<=) and CC(S,A,,<).

PROPOSITION D. Every A containedin [k]^k has a continuation S in
[t!]^k without (8k)!!-1, where C(S)^k, CC(S)^k are order equivalent
over 1!,2!,...,t!.

PROPOSITION E. Every A containedin [k]^k has a continuation S in
[t!]^k without (8k)!!-1, where C(S)^k, CC(S,A,<=)^k are order
equivalent over 1!,2!,...,t!.

PROPOSITION F. Every A containedin [k]^k has a continuation S in
[t!]^k without (8k)!!-1, where C(S)^k, CC(S,A,<)^k are order
equivalent over 1!,2!,...,t!.

D-F are explicitly Pi01.

THEOREM. Propositions D-F are provably equivalent to Con(SMAH) over ACA.

DEFINITION 5. Let A,B containedin {0,...,t}^k. A,B are similar below p
if and only if every x in of A union B is order equivalent to y in A
and z in B, where y,z have the same coordinates < p in the same

Here are somewhat simplified versions of

PROPOSITION 1. For all  t > (8k)!, every A containedin [k]^k has a
continuation S in [kt]^k, where C(S)^3[2t,3t,...,kt] and
CC(S)^3[t,2t,...,(k-1)t] are similar below t.

PROPOSITION 2. For all  t > (8k)!, every A containedin [k]^k has a
continuation in [kt]^k, where C(S)^3[2t,3t,...,kt] and
CC(S,A,<=)^3[t,2t,...,(k-1)t] are similar below t.

PROPOSITION 3. For all  t > (8k)!, every A containedin [k]^k has a
continuation in [kt]^k, where C(S)^3[2t,3t,...,kt] and
CC(S,A,<)^3[t,2t,...,(k-1)t] are similar below t.

Propositions 1-3 are explicitly Pi01.

THEOREM 4. Propositions 1-3 are provably equivalent to Con(SRP) over EFA.

My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
This is the 597th 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

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
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
593: Finite Continuation Theory 12/perfect?  8/23/15  9:47PM
594: Finite Continuation Theory 13/perfect?  8/4/15  1:44PM
595: Finite Continuation Theory 14/perfect?  8/5/15  8:23PM
596: Finite Continuation Theory 15/perfect?  8/8/15 12:35AM

Harvey Friedman

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