845: Brand New Tangible Incompleteness/3

[Harvey Friedman]

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

*INCOMPLETENESS IN THE RATIONALS*

DEFINITION 1. The upper shift/N is the map US/N:Q^k into Q^k where
US/N(x) is obtained by adding 1 to all coordinates greater than all
coordinates outside N

(In 843, 844 I was using "fractional coordinates" instead of
"coordinates outside N". This is the correct definition). .

BRAND NEW INCOMPLETENESS IN THE RATIONALS.. Every order invariant
subset of Q[-n,n]^2k has a completely US/N invariant maximal square.

*INCOMPLETENESS IN THE INTEGERS*

We now switch to the integers and use interval notation  in the integers.

DEFINITION 2. The upper shift/rN is the map US/rN:Q^k into Q^k where
US/tN(x) is obtained by adding r to all coordinates greater than all
coordinates outside rN.

We will only use US/rN on and into Z^k.

DEFINITION 3. Let S containedin Z^k and B containedin Z. The restriction
of S to B is the result of leaving all coordinates of elements of S
outside B blank.

E.g., the restriction of {(-1,2,0,4),(6.5.-2,0)} to 2N is
{(blank,2,0,4),(6,blank,blank,0)}.

DEFINITION 4. Let E containedin [s,t]^k and B containedin Z. S is a B
maximal square in E if and only if every square S' in E containing S has
the same restriction to B.

BRAND NEW INCOMPLETENESS IN THE INTEGERS.. Let r >> k. Every order
invariant subset of [-kr,kr]^2k has a completely US/rN invariant rN
maximal square.

The >> is rather innocent.

BRAND NEW INCOMPLETENESS IN THE INTEGERS.. Let r > (8k)! Every order
invariant subset of [-kr,kr]^2k has a completely US/rN invariant rN
maximal square.

Note that the first version is explicitly Pi03 whereas the second
version is explicitly Pi01.

THEOREM 2. Band New Incompleteness in the Integers (both forms) are
provably equivalent to Con(SRP) over EFA.

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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 845th 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

Harvey Friedman