[FOM] 564: Perfectly Natural Review #3

Mon Nov 24 01:19:23 EST 2014

THIS IS A REWORKING OF THE MAIN TEMPLATES FROM THE FIRST SECTION OF
http://www.cs.nyu.edu/pipermail/fom/2014-November/018406.html

We have reorganized the critical leadoff motivating Templates for this
entire line of investigation. Recall from
http://www.cs.nyu.edu/pipermail/fom/2014-November/018406.html

We now prefer to present these two critical motivating Templates first.

TEMPLATE A. Let E be an order theoretic equivalence relation on
Q[0,1]^k. Every order invariant relation on Q[0,1]^k has an E
invariant maximal root.

TEMPLATE B. Let E be an order theoretic equivalence relation on
Q[0,1]^k. For all order theoretic E_1,...,E_r isomorphic to E, every
order invariant relation on Q[0,1]^k has an E_1,...,E_r invariant
maximal root.

We know that large cardinals are necessary to completely analyze
Template A, and conjecture that they are necessary and sufficient. We
know that large cardinals are necessary and sufficient to completely
analyze Template B.

Here are two more templates using relations instead of equivalence
relations. We obtain the same results and make the same conjecture.

TEMPLATE C. Let P be an order theoretic relation on Q[0,1]^k. Every
order invariant relation on Q[0,1]^k has a maximal root containing its
P image.

TEMPLATE D.  Let P be an order theoretic relation on Q[0,1^k. For all
order theoretic P_1,...,P_r isomorphic to P, every order invariant
relation on Q[0,1]^k has a maximal root containing its P_1,...,P_r
images.

Here are another two using functions. We obtain the same results and
make the same conjecture.

TEMPLATE E. Let F:Q[0,1]^k into Q[0,1]^k be order theoretic. Every
order invariant relation on Q[0,1]^k has a maximal root containing its
F image.

TEMPLATE F. Let F:Q[0,1]^k into Q[0,1]^k be order theoretic. For all
order theoretic F_1,...,F_r isomorphic to F, every order invariant
relation on Q[0,1]^k has a maximal root containing its F_1,...,F_r
images.

We can also use relations on Q[0,1] and its images acting
coordinatewise. We obtain the same results and make the same
conjecture, - slightly modified to take into account that the
statements quantify over k.

TEMPLATE G. Let P_1,...,P_r be order theoretic relations on Q[0,1].
Every order invariant relation on Q[0,1]^k has a maximal root
containing its P_1,...,P_r images.

TEMPLATE H. Let P be an order theoretic relation on Q[0,1]. For all
order theoretic P_1,...,P_r isomorphic to P, every order invariant
relation on Q[0,1]^k has a maximal root containing its P_1,...,P_r
images.

We also can use one dimensional functions. We obtain the same results
and make the same conjecture as we did for relations on Q[0,1].

TEMPLATE I. Let f_1,...,f_r:Q[0,1] into Q[0,1] be order theoretic.
Every order invariant relation on Q[0,1]^k has a maximal root
containing its f_1,...,f_r images.

TEMPLATE J. Let f:Q[0,1] into Q[0,1] be order theoretic. For all order
theoretic f_1,...,f_r isomorphic to f, every order invariant relation
on Q[0,1]^k has a maximal root containing its f_1,...,f_r images.

If we restrict the above Templates to upward E's, P's, F's, f's, then
large cardinals are necessary and sufficient for complete
analyzability.

We still emphasize the use of sections and liftings under Spectifics.

In the statements involving the upper shift and reductions, as we have
seen before, large cardinals are generally necessary and sufficient to
analyze all of the relevant templates.

Although we now have a very good statement equivalent to Con(HUGE), in
Proposition 4.2 of
http://www.cs.nyu.edu/pipermail/fom/2014-November/018412.html , we
don't as of yet have an explicitly Pi01 statement equivalent to
Con(HUGE) that meets current standards. We are now working with a
sequential process that holds promise.

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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 564th 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

Harvey Friedman