# [FOM] 799: Beyond Perfectly Natural/5

Harvey Friedman hmflogic at gmail.com
Sun Apr 1 14:16:18 EDT 2018

```We have been in contact with a number of people across math logic,
various branches of mathematics, and computer science concerning
https://cs.nyu.edu/pipermail/fom/2018-March/020900.html

notion of "order equivalence" and also of the Complementation Theorem.

equivalence" and also some comments that have led us to finite tune
the presentation. So we have posted a fine tuned version at

102. A U. R<[A] - April 1, 2018

scientists, and will contact more of them. For math logicians, asking

COMMENT #!

The idea that two integer sequences are order-isomorphic is very
natural and occurs often in studying permutation statistics.

COMMENT #2

I’m OK with your order equivalence. I think your point is that order
invariant subsets can, however, somehow encode quite complex
combinatorics.

COMMENT #3

In a sense, yes, this notion is implicit in the development of
subsymmetric basic sequences and, especially, spreading models (a la
Brunel and Sucheston from the 1970s).  Ramsey's theorem is used for
the existence of spreading models, which are studied even now within
Banach space theory.

*************

#1 is a senior icon in combinatorial mathematics, #2 is a senior
Fields Medalist, and #3 is an unusually strong senior mathematician
focusing on Banach space theory. As far as math logic people are

The TCS situation is very puzzling, and I need to get to the bottom of
it. Perhaps I was unlucky enough to pick unrepresentative people from
TCS?

************************************************************************
My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
This is the 799th 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 #1-699 can be found at

700: Large Cardinals and Continuations/14  8/1/16  11:01AM
701: Extending Functions/1  8/10/16  10:02AM
702: Large Cardinals and Continuations/15  8/22/16  9:22PM
703: Large Cardinals and Continuations/16  8/26/16  12:03AM
704: Large Cardinals and Continuations/17  8/31/16  12:55AM
705: Large Cardinals and Continuations/18  8/31/16  11:47PM
706: Second Incompleteness/1  7/5/16  2:03AM
707: Second Incompleteness/2  9/8/16  3:37PM
708: Second Incompleteness/3  9/11/16  10:33PM
709: Large Cardinals and Continuations/19  9/13/16 4:17AM
710: Large Cardinals and Continuations/20  9/14/16  1:27AM
700: Large Cardinals and Continuations/14  8/1/16  11:01AM
701: Extending Functions/1  8/10/16  10:02AM
702: Large Cardinals and Continuations/15  8/22/16  9:22PM
703: Large Cardinals and Continuations/16  8/26/16  12:03AM
704: Large Cardinals and Continuations/17  8/31/16  12:55AM
705: Large Cardinals and Continuations/18  8/31/16  11:47PM
706: Second Incompleteness/1  7/5/16  2:03AM
707: Second Incompleteness/2  9/8/16  3:37PM
708: Second Incompleteness/3  9/11/16  10:33PM
709: Large Cardinals and Continuations/19  9/13/16 4:17AM
710: Large Cardinals and Continuations/20  9/14/16  1:27AM
711: Large Cardinals and Continuations/21  9/18/16 10:42AM
712: PA Incompleteness/1  9/23/16  1:20AM
713: Foundations of Geometry/1  9/24/16  2:09PM
714: Foundations of Geometry/2  9/25/16  10:26PM
715: Foundations of Geometry/3  9/27/16  1:08AM
716: Foundations of Geometry/4  9/27/16  10:25PM
717: Foundations of Geometry/5  9/30/16  12:16AM
718: Foundations of Geometry/6  101/16  12:19PM
719: Large Cardinals and Emulations/22
720: Foundations of Geometry/7  10/2/16  1:59PM
721: Large Cardinals and Emulations//23  10/4/16  2:35AM
722: Large Cardinals and Emulations/24  10/616  1:59AM
723: Philosophical Geometry/8  10/816  1:47AM
724: Philosophical Geometry/9  10/10/16  9:36AM
725: Philosophical Geometry/10  10/14/16  10:16PM
726: Philosophical Geometry/11  Oct 17 16:04:26 EDT 2016
727: Large Cardinals and Emulations/25  10/20/16  1:37PM
728: Philosophical Geometry/12  10/24/16  3:35PM
729: Consistency of Mathematics/1  10/25/16  1:25PM
730: Consistency of Mathematics/2  11/17/16  9:50PM
731: Large Cardinals and Emulations/26  11/21/16  5:40PM
732: Large Cardinals and Emulations/27  11/28/16  1:31AM
733: Large Cardinals and Emulations/28  12/6/16  1AM
734: Large Cardinals and Emulations/29  12/8/16  2:53PM
735: Philosophical Geometry/13  12/19/16  4:24PM
736: Philosophical Geometry/14  12/20/16  12:43PM
737: Philosophical Geometry/15  12/22/16  3:24PM
738: Philosophical Geometry/16  12/27/16  6:54PM
739: Philosophical Geometry/17  1/2/17  11:50PM
740: Philosophy of Incompleteness/2  1/7/16  8:33AM
741: Philosophy of Incompleteness/3  1/7/16  1:18PM
742: Philosophy of Incompleteness/4  1/8/16 3:45AM
743: Philosophy of Incompleteness/5  1/9/16  2:32PM
744: Philosophy of Incompleteness/6  1/10/16  1/10/16  12:15AM
745: Philosophy of Incompleteness/7  1/11/16  12:40AM
746: Philosophy of Incompleteness/8  1/12/17  3:54PM
747: PA Incompleteness/2  2/3/17 12:07PM
748: Large Cardinals and Emulations/30  2/15/17  2:19AM
749: Large Cardinals and Emulations/31  2/15/17  2:19AM
750: Large Cardinals and Emulations/32  2/15/17  2:20AM
751: Large Cardinals and Emulations/33  2/17/17 12:52AM
752: Emulation Theory for Pure Math/1  3/14/17  12:57AM
753: Emulation Theory for Math Logic  3/10/17  2:17AM
754: Large Cardinals and Emulations/34  3/12/17  12:34AM
755: Large Cardinals and Emulations/35  3/12/17  12:33AM
756: Large Cardinals and Emulations/36  3/24/17  8:03AM
757: Large Cardinals and Emulations/37  3/27/17  2:39AM
758: Large Cardinals and Emulations/38  4/10/17  1:11AM
759: Large Cardinals and Emulations/39  4/10/17  1:11AM
760: Large Cardinals and Emulations/40  4/13/17  11:53PM
761: Large Cardinals and Emulations/41  4/15/17  4:54PM
762: Baby Emulation Theory/Expositional  4/17/17  1:23AM
763: Large Cardinals and Emulations/42  5/817  2:18AM
764: Large Cardinals and Emulations/43  5/11/17  12:26AM
765: Large Cardinals and Emulations/44  5/14/17  6:03PM
766: Large Cardinals and Emulations/45  7/2/17  1:22PM
767: Impossible Counting 1  9/2/17  8:28AM
768: Theory Completions  9/4/17  9:13PM
769: Complexity of Integers 1  9/7/17  12:30AM
770: Algorithmic Unsolvability 1  10/13/17  1:55PM
771: Algorithmic Unsolvability 2  10/18/17  10/15/17  10:14PM
772: Algorithmic Unsolvability 3  Oct 19 02:41:32 EDT 2017
773: Goedel's Second: Proofs/1  Dec 18 20:31:25 EST 2017
774: Goedel's Second: Proofs/2  Dec 18 20:36:04 EST 2017
775: Goedel's Second: Proofs/3  Dec 19 00:48:45 EST 2017
776: Logically Natural Examples 1  12/21 01:00:40 EST 2017
777: Goedel's Second: Proofs/4  12/28/17  8:02PM
778: Goedel's Second: Proofs/5  12/30/17  2:40AM
779: End of Year Claims  12/31/17  8:03PM
780: One Dimensional Incompleteness/1  1/4/18  1:14AM
781: One Dimensional Incompleteness/2  1/6/18  11:25PM
782: Revolutionary Possibilities/1  1/12/18  11:26AM
783: Revolutionary Possibilities/2  1/20/18  9:43PM
784: Revolutionary Possibilities/3  1/21/18  2:59PM
785: Revolutionary Possibilities/4  1/22/18  12:38AM
786: Revolutionary Possibilities/5  1/24/18  12:15AM
787: Revolutionary Possibilities/6  1/25/18  4:09AM
788: Revolutionary Possibilities/7  2/1/18  2:18AM
789: Revolutionary Possibilities/8  2/1/18  9:02AM
790: Revolutionary Possibilities/9  2/2/18  3:07AM
791: Emulation Theory/Inductive Equations/1  2/8/18  11:52PM
792: Emulation Theory/Inductive Equations/2  2/25/18  12:58PM
793: Emulation Theory/Inductive Equations/3  2/27/18  12:11AM
794: Emulation Theory/Inductive Equations/4  3/6/18  9:22PM
795: Beyond Perfectly Natural/1  3/19/18  7:19PM
796: Beyond Perfectly Natural/2  3/24/18  11:29PM
797: Beyond Perfectly Natural/3  3/27/18  12:37AM
798: Beyond Perfectly Natural/4  3/31/18  5:30PM

Harvey Friedman
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