[FOM] 830: Tangible Incompleteness Restarted/4

Thu Sep 26 13:17:36 EDT 2019

NOTE. In the previous #3, the table of contents did not correspond
exactly to the contents. Below see a fixed TOC, with the section 2.1.2
title modified.

Here we add material to 2.1.2 and add sections 2.1.4 and 2.2.3.  The
new material concerns the use of other N shifts - other than our N
tail shift.

NOTE: The plan is to fully incorporate the previous

TANGIBLE MATHEMATICAL INCOMPLETENESS OF ZFC
https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
August 16, 2018, #106, 66 pages

into the present development here. This 8/18 paper is based on a very
weak notion of stability. As indicated previously, we now prefer to
use the strongest notions of stability. We postpone working out the
technical details in the very weak notions of stability in favor of
working out the details with the strongest ones. The strongest ones
have the advantage in that they make it rather promising to show that
incompleteness from ZFC occurs at very low dimension, say 3 or 4.

This 8/18 paper does not touch Logical Classes which we are about to
take up here planned not for the present #4 but for #5.

This 8/18 paper is very detailed and keeps track of a lot of
parameters in the statements that we are not dealing with here. In the
combined paper we will continue to operate at that level of detail.
However, we will work hard to make things very readable. For example,
we will tart off by presenting and discussing versions where we do not
keep track of any parameters (like we are doing here). We did some of
that already in 8/18 but we can do better along these lines.

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 - here
   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. - here
   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

################

2.1.2. N (SUB)TAIL, N (SUB)TAIL SHIFT, N TAIL-RELATED

Continued from #3. It is better to write N tail shift-related on Q^k
rather than N tail shift related on Q^k, for that important binary
relation on Q^k, to avoid confusion.

DEFINITION 2.1.2.4. An N subtail of x in Q^k is a set A of coordinates
of x such that if x_j >= x_i in A then x_j in A intersect N. Subtails
are held in place. An N subtail shift on Q^k is an h:Q^k into Q^k such
that each h(x) results in adding 1 to some N subtail of x.

The following relates the equivalence relation N tail shift-related to shifting.

THEOREM 2.1.2.1. N tail shift-related on Q^k is the least equivalence
relation on Q^k that contains the graph of all N subtail shifts on
Q^k.

2.1.4. EMULATION USABILITY OF SHIFTS

DEFINITION 2.1.4.1. A shift on Q^k is an h:Q^k into Q^k such that each
f(x) is obtained by adding 1 from zero to all coordinates of x. x,y in
Q^k are N order equivalent if and only if x,y are order equivalent and
for all i <= k, x_i in N iff y_i in N. A containedin Q^k is N order
invariant if and only if for all N order equivalent x,y in Q^k, x in A
iff y in A. h:Q^k into Q^k is a uniform shift if and only if h is a
shift on Q^k such that for all N order equivalent x,y in Q^k, h(x)-x =
h(y)-y.

DEFINITION 2.1.4.2. A shift h:Q^k into Q^k is emulation usable if and
only if the following holds. Every finite E containedin Q[-n,n]^k has
a completely h invariant maximal emulator.

PROPOSITION 2.1.4.1. Let h be a uniform shift on Q^k. The following
are equivalent.
i. h is emulation usable.
ii. h is an N subtail shift.
iii. h is contained in N tail-related on Q^k.
Furthermore, N tail-related on Q^k is the union of all emulation
usable uniform shifts on N^k.

THEOREM 2.1.4.2. Proposition 2.1.4.1 (both parts) is provably
equivalent to Con(SRP) over WKL_0.

2..2.3. CLIQUE USABILITY OF SHIFTS

DEFINITION 2.2.3.1. h:Q^k into Q^k is clique usable if and only if the
following holds. Every order invariant graph on Q[-n,n]^k has a
completely  h invariant maximal clique.

PROPOSITION 2.2.3.1. Let h be a uniform shift on Q^k. The following
are equivalent.
i. h is clique usable.
ii. h is an N subtail shift.
iii. h is contained in N tail-related on Q^k.
Furthermore, N tail-related on Q^k is the union of all clique usable
uniform shifts on N^k.

THEOREM 2.2.3.2. Proposition 2.2.3.1 (both parts) is provably
equivalent to Con(SRP) over WKL_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 830th 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

Harvey Friedman