[FOM] 826: Tangible Incompleteness/5

[Harvey Friedman]

BACK ON TRACK!

I had recently replaced the original bare bones extremely weak looking
notion of stability with a stronger one, still quite good - because I
didn't think I would be able to reverse with the original weak one.

HOWEVER, I am now back on track to finishing the manuscript reversing
the original stability, which is on the Q[0,k]^k (not the more general
Q[0,n]^k). I see how to work with the original stability. This is

1) S containedin Q[0,k]^k is stable if and only if for all p < 1,
(p,1,...,k-1) in S iff (p,2,...,k) in S.

Now  the three lead statements for Emulation Theory are:

MAXIMAL EMULATION STABILITY. MES. Every subset of Q[0,k]^k has a
stable maximal emulator.

MAXIMAL DUPLICATION STABILITY. MDS. Every subset of Q[0,k]^k has a
stable maximal duplicator.

MAXIMAL CLIQUE STABILITY. MCS. Every order invariant graph on Q[0,k]^k
has a stable maximal clique.

I am nearly finished with a polished reworking of

http://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
   #106

It should be on that site in a few days, as #108.

Also I give a complete characterization of all of the notions of
stability that can be used with MES, MDS, MCS, where of course our
default stability 1) is a special case. I am right in the middle of
the writeup of that proof, right now.

I define the three very natural coordinate free order invariant/Z
equivalence relations DZ[k], full upZ[k], full dwnZ[k]. In #108 I
prove

CHARACTERIZATION. A coordinate free order invariant/Z equivalence
relation on Q^k can be used for stability in MES if and only if it is
included in at least one of DZ[k], full upZ[k], full down[k].

THEOREM. This Characterization is provably equivalent to Con(SRP) over WKL_0.

Also #108 will have a complete proof that MES MDS, MCS are provably
equivalent to Con(SRP) over WKL_0, relying on the previous Putam
volume paper, to appear.  The Putnam Volume paper is at

http://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
   #92

The missing major result of Emulation Theory is of course the promised
reversal, showing that MES, MDS, MCS each imply Con(SRP) over RCA_0.

The reversal paper is going quite well, but was halted in order to
finish #108, which is nearing completion.

I have a self imposed deadline of 9/23/18 for the completion of #108
and the just mentioned Reversal. Lots of refinements, including the
using of very small k,n in Q[0,n]^k to get independence from ZFC, may
have to take longer.

************************************************************************
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 826th 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

Harvey Friedman