# 846: Brand New Tangible Incompleteness/4

Harvey Friedman hmflogic at gmail.com
Wed Apr 1 00:32:02 EDT 2020

```IS THIS FINALLY THE PINNACLE OF THE 53-YEAR OLD TANGIBLE
INCOMPLETENESS PROJECT IN TERMS OF RAW COMBINATORIAL SIMPLICITY?
\$\$\$\$\$\$\$\$\$\$\$\$\$\$

I just put up a revised version of #110 in my Downloadable Manuscripts

This is considerably more polished than 843, 844, 845 here.

INVARIANT MAXIMAL SQUARES. Every order invariant subset of every
Q[-n,n]^k has a complete USH/N invariant maximal square.

We use the weak form of maximality called N maximality discussed here,
but defined better in #110

INVARIANT N MAXIMAL SQUARES. Every order invariant subset of every
Q[-n,n]^k has a complete USH/N invariant finite N maximal square.

This is explicitly finite, in fact explicitly Pi02. It becomes
explicitly Pi01 via an a priori exponential upper bound on the size of
the finite N maximal square.

Then this is naturally taken into the integers using >>.

INTEGER INVARIANT MAXIMAL SQUARES/>>. Let t >> k. Every order
invariant subset of every [-kt,kt]^k has a completely USH/tN invariant
tN maximal square.

This is explicitly Pi03 because of the >>. This is trivially fixed to:

INTEGER INVARIANT MAXIMAL SQUARES. Let t > (8k)!. Every order
invariant subset of every [-kt,kt]^k has a completely USH/tN invariant
tN maximal square.

And the above is explicitly Pi01.

All of these statements are provably equivalent to Con(SRP) over WKL0.

AGENDA:

1. Write exactly #110 up with complete proofs of everything. Show
Q[-3,3]^3 is enuf.
2. Revise and put on website my version for Gifted High School.
3. Higher Cardinals going to n-Huge. Greatly improve on earlier work.
4. Incorporate more mathematical structure. The present is incredibly
elemental.

#######################################

My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
This is the 846th 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 #1-799 can be found at

800: Beyond Perfectly Natural/6  4/3/18  8:37PM
801: Big Foundational Issues/1  4/4/18  12:15AM
802: Systematic f.o.m./1  4/4/18  1:06AM
803: Perfectly Natural/7  4/11/18  1:02AM
804: Beyond Perfectly Natural/8  4/12/18  11:23PM
805: Beyond Perfectly Natural/9  4/20/18  10:47PM
806: Beyond Perfectly Natural/10  4/22/18  9:06PM
807: Beyond Perfectly Natural/11  4/29/18  9:19PM
808: Big Foundational Issues/2  5/1/18  12:24AM
809: Goedel's Second Reworked/1  5/20/18  3:47PM
810: Goedel's Second Reworked/2  5/23/18  10:59AM
811: Big Foundational Issues/3  5/23/18  10:06PM
812: Goedel's Second Reworked/3  5/24/18  9:57AM
813: Beyond Perfectly Natural/12  05/29/18  6:22AM
814: Beyond Perfectly Natural/13  6/3/18  2:05PM
815: Beyond Perfectly Natural/14  6/5/18  9:41PM
816: Beyond Perfectly Natural/15  6/8/18  1:20AM
817: Beyond Perfectly Natural/16  Jun 13 01:08:40
818: Beyond Perfectly Natural/17  6/13/18  4:16PM
819: Sugared ZFC Formalization/1  6/13/18  6:42PM
820: Sugared ZFC Formalization/2  6/14/18  6:45PM
821: Beyond Perfectly Natural/18  6/17/18  1:11AM
822: Tangible Incompleteness/1  7/14/18  10:56PM
823: Tangible Incompleteness/2  7/17/18  10:54PM
824: Tangible Incompleteness/3  7/18/18  11:13PM
825: Tangible Incompleteness/4  7/20/18  12:37AM
826: Tangible Incompleteness/5  7/26/18  11:37PM
827: Tangible Incompleteness Restarted/1  9/23/19  11:19PM
828: Tangible Incompleteness Restarted/2  9/23/19  11:19PM
829: Tangible Incompleteness Restarted/3  9/23/19  11:20PM
830: Tangible Incompleteness Restarted/4  9/26/19  1:17 PM
831: Tangible Incompleteness Restarted/5  9/29/19  2:54AM
832: Tangible Incompleteness Restarted/6  10/2/19  1:15PM
833: Tangible Incompleteness Restarted/7  10/5/19  2:34PM
834: Tangible Incompleteness Restarted/8  10/10/19  5:02PM
835: Tangible Incompleteness Restarted/9  10/13/19  4:50AM
836: Tangible Incompleteness Restarted/10  10/14/19  12:34PM
837: Tangible Incompleteness Restarted/11 10/18/20  02:58AM
838: New Tangible Incompleteness/1 1/11/20 1:04PM
839: New Tangible Incompleteness/2 1/13/20 1:10 PM
840: New Tangible Incompleteness/3 1/14/20 4:50PM
841: New Tangible Incompleteness/4 1/15/20 1:58PM
842: Gromov's "most powerful language" and set theory  2/8/20  2:53AM
843: Brand New Tangible Incompleteness/1 3/22/20 10:50PM
844: Brand New Tangible Incompleteness/2 3/24/20  12:37AM
845: Brand New Tangible Incompleteness/3 3/28/20 7:25AM

Harvey Friedman
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