[FOM] 571: Philosophy of Incompleteness 2

Thu Jan 8 11:30:13 EST 2015

Here I will briefly jump to my Perfectly Mathematically Natural
Concrete Incompleteness from the usual ZFC axioms for mathematics.

We have made these More Perfect, and expect to make then Yet More Perfect.

From

[1] https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
#87, Perfectly Mathematically Natural Concrete Incompleteness,  -
order theoretic relations. December 14, 2014, 25 pages. Extended
abstract.

we have, quoting from the Abstract,

1) every order invariant relation on Q[0,1]^k has a maximal root whose
projections at tuples from {1,1/2,...,1/n}^r> agree below 1/n,

2) every order invariant relation on Q^k has a # basis containing its
upper shift.

First, with regard to 1). I see two ways to make this arguably more striking.

One is to use the usual notion of subsequence, rather than the exponent r>.

PROPOSITION A. Every order invariant relation on Q[0,1]^k has a
maximal root whose projections at length r subsequences of
(1,1/2,...,1/n) agree below 1/n.

The second is to use very small numbers k,n,r. Now just how small I
can make k,n,r here is not going to be confidently determined until I
am right smack in the middle of writing up the reversal.

But let me state a really good seemingly realistic but ambitious target:

PROPOSITION B. (target). Every order invariant relation on Q[0,1]^4
has a maximal root whose projections at length 3 subsequences of
(1,1/2,1/3,1/4) agree below 1/4.

Obviously, I might fail to reverse Proposition B, but should be able
to reverse it with numbers a little bit bigger. We shall see.

Now coming to 2), there is an annoying thing in the definition of #
basis in [1] above. Recall that I had to define S#, S containedin Q^k,
as the least E^k containing S union {0}^k.

I would rather stick with the most natural S#, which is simply the
least E^k containing S. Then the origin will surface again in the
conclusion. This is perfectly natural, as one is going to have to
disallow S = emptyset no matter what. Now, using this most natural
notion of #, the statement

STATEMENT. Every order invariant relation on Q^k has a # basis
containing its upper shift

becomes trivial, taking the empty # basis. On the other hand,

STATEMENT. Every order invariant relation on Q^k has a # basis
containing its upper shift and the origin.

is easily refutable using the the order invariant relation Q^2k.

PROPOSITION C. Every order invariant relation on Q^k not involving the
origin has a # basis containing its upper shift and the origin.

I like this better. There is always a danger that I like something new
better than anything old, and so have to be careful. But here I think
that the cost of making # basis as pristine as possible is well worth
it. After all, the definition of # basis is the only place subject to
any complaint.

PROPOSITION D. (target). Every order invariant relation on Q^4 not
involving the origin has a # basis containing its upper shift and the
origin.

Proposition B should correspond to a weak fragment of SRP, but
Proposition D should correspond to all of SRP.

I will be upgrading [1] with A-D above and also with
http://www.cs.nyu.edu/pipermail/fom/2015-January/018489.html shortly.

************************************************************
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 571st 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

Harvey Friedman