[FOM] 825: Tangible Incompleteness/4

[Harvey Friedman]

I mangled a little bit the Characterization in section 3 of
https://cs.nyu.edu/pipermail/fom/2018-July/021108.html

I wrote

SYMMETRIC ORDER INVARIANT/Z ME USE. SOIZMEU. A symmetric order
invariant/Z equivalence relation on Q^k is ME usable if and only if it
is contained in full upZ, contained in full downZ, or contained in the
Z-diagonal {((p,...,p),(q,...,q)): p,q in Z or p = q}.

What I meant to write was:

SYMMETRIC ORDER INVARIANT/Z ME USE. SOIZMEU. A symmetric order
invariant/Z equivalence relation on Q^k is ME usable if and only if it
is contained in full upZ, contained in full downZ, or contained in
{(x,y): x = y in Q^k or x,y in Z^k are order equivalent}.

I am putting together manuscript 108 on my Downloadable page which
will prove this characterization.

The above is somewhat cautious, and I am going for the stronger
result: remove "symmetric" from the above statement.

REMARKS. There are a couple of remarks that will appear in 108.

1. There is a fixed finite subset of some fixed Q[0,n]^k such that
"there exists a stable maximal emulator" is independent of ZFC. I
think that k,n can be made very small, but the finite subset of
Q[0,n]^k not small and not pretty.

2. In MES, MDS, MCS, the maximal emulator, maximal duplicator, maximal
clique can be found recursive in the halting problem.

************************************************************************
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This is the 825th 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

Harvey Friedman