[FOM] 831: Tangible Incompleteness Restarted/5

Sun Sep 29 02:54:48 EDT 2019

Before continuing with sections 2.3, 2.4, 3, we add some material to
sections 2.1, 2.2. We won't worry here about changing the numberings
and the exact position of the new material. We will just focus on the
new material. First we repeat the TOC for the convenience of the
reader.

TANGIBLE INCOMPLETENESS

1. BOOLEAN RELATION THEORY - my website

2. INVARIANT MAXIMALITY
   2.1. Emulation Theory - #3
      2.1.1. N, Z+, Q, Q[(a,b)], order equivalent, maximal emulator,
invariance - #3
      2.1.2. N (sub)tail, N (sub)tail shift, N tail-related - #3
      2.1.3. Invariant Maximal Emulation - #3
      2.1.4. Emulation usability of shifts - #4
   2.2. Invariant Graph Theory - #3
      2.2.1. Graphs, order invariant graphs, cliques - #3
      2.2.2. Invariant Maximal Cliques - #3
      2.2.3. Clique usability of shifts. - #4
   2.3. A...,A Classes - later
      2.3.1. (Q,<,rels) - later
      2.3.2. (D,<,rels), D containedin Q - later
      2.3.3. (D,<,fcns), D containedin Q - later
      2.3.4. (D,<,...), D containedin Q - later
   2.4. A...AE...E Clases - later
      2.4.1. (Q,<,rels) - later
      2.4.2. (D,<,rels), D containedin Q - later
      2.4.3. (D,<,fans), D containedin Q - later
      2.4.4. (D,<,...), D containedin Q - later

3. SEQUENTIAL CONSTRUCTIONS - in development

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We have seen that N Shift-Related is the really natural equivalence
relation on Q^k that drives our invariance. We have seen that it is
maximal for being emulation and clique usable, and also is the least
equivalence relation containing all of the N subtail shifts. I am
still trying to get a stronger theorem to the effect that N
Shift-Related on Q^k is the maximum equivalence relation on Q^k
satisfying its own fundamental invariance conditions (as a set of
2k-tuples) which is emulation or clique usable. There are some
technical issues left here.

So we would like a name for N Shift-Related invariance. We will use
the word "stable". We have used it before for a very weak case of
this. So using the word "stable" we have the following lead statements
of Invariant Maximality (before we consider logical classes in section
2.3).

IME. Invariant Maximal Emulation. Every finite subset of Q[-n,n]^k has
a stable maximal emulatior.

IMC. Invariant Maximal Clique. Every order invariant graph on
Q[-n,n]^k has a stable maximal clique.

IME and IMC are provably equivalent to Con(SRP) over WKL_0 with the
forward direction over RCA_0. In dimension 2 IME and IMC are already
substantial, and the natural proofs correspond to Con(Z_2), even for
Q[-1,1]^2. However, we believe that RCA_0 suffices through a
painstaking classification of the finite number of order invariant
graphs on Q[-1,1]^2.

Q[-n,n]^2 seem perfect for the Magic Bullet from
https://cs.nyu.edu/pipermail/fom/2019-September/021689.html Perhaps
especially Q[-1,1]^2 and Q[-2,2]^2, with the more advanced Q[-2,2]^3
and Q[-3,3]^3 for the brave. We don't have a proof for Q[-2,2]^3
without using a substantial large cardinal.

We now introduce two elaborations of IME and IMC. The first is already
present in

Tangible Mathematical Incompleteness of ZFC
https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
August 16, 2018, #106, 66 pages

whereas the second is not.

DEFINITION A. S is an r-emulatior of E containedin Q[-n,n]^k if and
only if every element of S^r is order equivalent to some element of
E^r.

Note that emulators are 2-emulators.

IMHE. Invariant Maximal Hyper Emulation. Every finite subset of
Q[-n,n]^k has a stable maximal r-emulatior.

IMHC. Invariant Maximal Hyper Clique. Every order invariant r-graph on
Q[-n,n]^k has a stable maximal r-clique.

3-emulation in Q[-1,1]^2 should open up a new world for the Magic Bullet.

The new elaboration is what we call tower emulation and tower clique. .

DEFINITION B. A maximal r-emulatior of E_1 containedin ... containedin
E_m containedin Q^k is an S_1 containedin ... containedin S_m
containedin Q^k such that each S_i is a maximal r-emulator of E_i.

IMTE. Invariant Maximal Tower Emulation. Every finite tower of subsets
of Q^k has a stable maximal emulator.

IMTC. Invariant Maximal Tower Clique. Every finite tower of order
invariant graphs on Q^k has a stable maximal clique.

IMTHE. Invariant Maximal Tower Hyper Emulation. Every finite tower of
subsets of Q^k has a stable maximal r-emulator.

IMTHC. Invariant Maximal Tower Hyper Clique. Every finite tower of
order invariant r-graphs on Q^k has a stable maximal r-clique.

The towers are so powerful that we no longer need the right endpoint
in Q[-n,n], and can use Q.

All of these statements are provably equivalent to Con(SRP) over
WKL_0, with the forward direction over RCA_0.

************************************************************************
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 831st 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

Harvey Friedman