# [FOM] 595: Finite Continuation Theory 14/perfect?

Harvey Friedman hmflogic at gmail.com
Wed Aug 5 20:23:44 EDT 2015

```In http://www.cs.nyu.edu/pipermail/fom/2015-August/018866.html I
presented three statements on the reals:

PROPOSITION A (reals). Every finite subset of R^k|>n has a maximal nonnegative
continuation, where S_1...n|>n = S_0...n-1|>n.

PROPOSITION B (reals). Let E be an equivalence relation on R^k
with finitely many equivalence classes. Every finite subset of R^k|>n
has a finite E-maximal nonnegative continuation, where S_1...n|>n =
S_0...n-1|>n.

PROPOSITION C (reals). Let E be an order theoretic equivalence relation on R^k
with finitely many equivalence classes. Every finite subset of R^k|>n
has a finite E-maximal nonnegative continuation, where S_1...n|>n =
S_0...n-1|>n.

and claimed that all three are provably equivalent to Con(SRP) over
ZC, Z, RCA0, respectively.

I quoted the upward Skolem Lowenheim for the proof of Proposition A (reals).

HOWEVER, that was a beginning student kind of error. You only get
Proposition A (reals) for some dense linear ordering of cardinality c
that way. Not for the reals.

This error only applies to A (reals) and not to B,C (reals).

Bottom line is that I believe Proposition A (reals) is refutable in Z.
There are some problems putting together a correct refutation. I stand
by what I wrote about Propositions B,C (reals) - equivalent to
Con(SRP).

I now want to take up a natural strenghening of Continuation Theory.
With this strengthening, the proofs from large cardinals require an
additional satisfying twist. The reversals get somewhat easier.

Note that in continuations, we consider only order types of
concatenated pairs of elements. Each concatenated pair from the
continuation is required to be order equivalent to some concatenated
pair from the set being continued.

We can consider our continuations as, therefore, 2-continuations. Here
2 means that we are concatenating merely ordered pairs of the vector
elements. More generally, for r-continuations, we concatenate r tuples
of vector elements.

So now we have the following six statements:

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

PROPOSITION 2 Let E be an equivalence relation on Q^k
with finitely many equivalence classes. Every finite subset of Q^k|>n
has a finite E-maximal nonnegative r-continuation, where S_1...n|>n =
S_0...n-1|>n.

PROPOSITION 3. Let E be an order theoretic equivalence relation on Q^k
with finitely many equivalence classes. Every finite subset of Q^k|>n
has a finite E-maximal nonnegative r-continuation, where S_1...n|>n =
S_0...n-1|>n.

PROPOSITION 4. Every finite subset of R^k|>n has a maximal nonnegative
r-continuation, where S_1...n|>n = S_0...n-1|>n.

PROPOSITION 5. Let E be an equivalence relation on R^k
with finitely many equivalence classes. Every finite subset of R^k|>n
has a finite E-maximal nonnegative r-continuation, where S_1...n|>n =
S_0...n-1|>n.

PROPOSITION 6. Let E be an order theoretic equivalence relation on R^k
with finitely many equivalence classes. Every finite subset of R^k|>n
has a finite E-maximal nonnegative r-continuation, where S_1...n|>n =
S_0...n-1|>n.

THEOREM 7. All but Proposition 4 are provably equivalent to Con(SRP).

As for Proposition A (reals), I believe that Proposition 4 is
refutable in Z, as it implies the former.

There is a further important strengthening. Instead of continuing a
single finite set, we continue a tower of finite sets. Rather than
treat this as an independent strengthening of continuations, we will
combine these by using r-continuations.

DEFINITION 1. Let W be a tower of subsets of Q^k|>=0 of length <=
omega. A nonnegative r-continuation tower of W is a tower of
respective r-continuations. A maximal nonnegative r-continuation tower
of W is a tower of respective maximal nonnegative r-continuations.

THEOREM 8. Every tower of subsets of Q^k|>=0 has a maximal nonnegative
r-continuation tower.

PROPOSITION 9. Every finite length tower of finite subsets of Q^k|>n
has a maximal nonnegative r-continuation tower whose components obey
S_1...n|>n = S_0...n-1|>n.

PROPOSITION 10. Every tower of finite subsets of Q^k|>n has a maximal
nonnegative continuation tower whose components obey S_1...n|>n =
S_0...n-1|>n.

Propositions 9.10 are implicitly Pi01 via Goedel's Completeness Theorem.

PROPOSITION 11. Let E be an order theoretic equivalence relation on
Q^k|>=0 with finitely many equivalence classes. Every finite length
tower of finite subsets of Q^l|>n has an E-maximal nonnegative
continuation tower of finite sets whose components obey S_1...n|>n =
S_0...n-1|>n.

PROPOSITION 12. Let E be an order theoretic equivalence relation on
R^k|>=0 with finitely many equivalence classes. Every finite length
tower of finite subsets of R^l|>n has an E-maximal nonnegative
continuation tower of finite sets whose components obey S_1...n|>n =
S_0...n-1|>n.

THEOREM 13. Propositions 9-12 are provably equivalent to Con(SRP) over WKL0.

************************************************************
My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
This is the 595th 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
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

Harvey Friedman
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