843: Brand New Tangible Incompleteness/1

Sun Mar 22 22:50:53 EDT 2020

There have been several breakthrough ideas in various directions since
the New Tangible Incompleteness/1-4 series, 1/11/20 - 1/15/20.

There is a very natural mapping from Q^k into Q^k called the upper N
shift, UNS:Q^k into Q^k. UNS(x) is obtained by adding 1 to all
nonnegative integer coordinates of x that are greater than all
fractional coordinates of x.

We have thought for some time about using UNS for Tangible
Incompleteness. However, it did not seem to be rich enough to really
work for the reversals. HOWEVER, now we see how to use it more
strongly.

We use UNS in the following way: S containedin Q[-n,n]^k is completely
UNS invariant if and only if for all x,UNS(x) in Q[-n,n]^k, x in S iff
UNS(x) in S.

Note that here we are using Q[-n,n]^k as the ambient space for S.

BRAND NEW. Every order invariant subset of Q[-n,n]^2k contains a
completely UNS invariant maximal square.

THEOREM. Brand New is implicitly Pi01 via the Goedel Completeness
Theorem. Brand New is provably equivalent to Con(SRP) over WKL_0.

Now for the explicitly Pi01 form.

Let S containedin Q^k The N trace of S is obtained from S by replacing
all coordinates of elements of S that are not in N by -1.

We say that S is a weakly maximal square in E containedin Q[-n,n]^2k
if and only if there is no square S' in E containing S with a more
inclusive N trace than S.

BRAND NEW FINITE. Every order invariant subset of Q[-n,n]^2k contains
a completely UNS invariant finite weakly maximal square.

THEOREM. Brand New Finite is provably equivalent to Con(SRP) over RCA_0.

There is an easy a priori upper bound on the size of the finite
square. This combined with quantifier elimination for (Q,<) puts Brand
New Finite in explicitly Pi01 form.

GIFTED HIGH SCHOOL

For gifted high school we still use emulators and avoid maximal
squares. Maximal emulators in two dimensions are special cases of
maximal squares in four dimensions where the side is in two
dimensions. So we use maximal emulators in two dimensions for gifted
high school.

For gifted high school we mostly use the space Q[-1,1]^2. Invariance
with respect to UNS in this space can be given by the three clauses

1. (0,0) in S iff (1,1) in S.
2. For p < 0, (p,0) in S iff (p,1) in S.
3. For p < 0, (0,p) in S iff (1,p) in S.

The gifted high schoolers can now pretty well understand this form of
tangible incompleteness:

BRAND NEW HIGH SCHOOL. Every finite subset of Q[-n,n]^k has a
completely UNS invariant maximal emulator.

UNS invariance can be understood by at least the better of the gifted
high schoolers.

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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 843th 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

Harvey Friedman