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
page https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
This is considerably more polished than 843, 844, 845 here.
The lead statement is
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
https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
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
https://www.youtube.com/channel/UCdRdeExwKiWndBl4YOxBTEQ
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
http://u.osu.edu/friedman.8/foundational-adventures/fom-email-list/
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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