[FOM] 833: Tangible Incompleteness Restarted/7

Harvey Friedman hmflogic at gmail.com
Sat Oct 5 14:34:54 EDT 2019

We have made a major advance in justifying the invariance that we use
for our lead statements. This matter was addressed pretty well in
sections 2.1.4 and 2.2.3, which appears in #4. We can now address it
considerably better. We have put the following paper on my website:

[1] 108. Tangible Incompleteness, Interim Report, October 6, 2019.

This [1] covers exactly the present state of section 2 in our TOC,
which is titled Invariant Maximality. In the next posting in this
series, #8, we plan to move on to section 3, Inductive Domains.

NOTE: So far we have been focusing on Combinatorial Invariance
properties. We will add later some material on Geometric Invariance
properties. Like isometries between line segments.We started this
investigation in 2018.

Let us recall some history. The lead statement in Emulation theory
reads, informally,

IME. Invariant Maximal Emulation. Every finite subset of Q[-n,n]^k has
an INVARIANT maximal emulator.

The maximal emulator notion is exquisite. The big question is: what is

It is absolutely crucial for INVARIANCE to be

1. immediately exquisite - like maximal emulator; OR
2. be reasonably simple, transparent, and natural AND naturally
identifiable among all invariance notions that are usable in IME.

Before we recently restarted Tangible Incompleteness (this year), we
have been persistently struggling with 1, with the idea of making the
notion of Invariance as WEAK as possible. This minimalistic approach
has run into two difficulties. Firstly, there is a tension between
making it weak and making it exquisitely natural. Secondly, sorting
out the usability of the relevant weak notions of invariance for
achieving UNPROVABILITY in ZFC, leads to investigations of HORRIFIC
COMPLEXITY. These complexities are of course rather mathematically
delicious down the road, maybe forming the basis of 100 Ph.D. theses
with incredibly brilliant ideas, but VERY BAD for the founding of a
new subject.

Since Restarting, I have decided to go for making the notion of
invariance as STRONG as possible. This began first with making some
minor improvements in the direct definitions of INVARIANCE. As we have
seen, we have arrived at

IME. Invariant Maximal Emulation. Every finite subset of Q[-n,n]^k has
a completely N tail invariant maximal emulator.

IME. Invariant Maximal Emulation. Every finite subset of Q[-n,n]^k has
a N tail-related invariant maximal emulator.

"Completely N tail invariant" is a bit simpler to explain than "N
tail-related invariant". We proved that both notions are equivalent.
And we called them STABILITY:

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

This would be more than natural enough for almost all mathematical
purposes. HOWEVER, this is Incompleteness and Foundations of
Mathematics, and so the standards for naturalness are INCOMPARABLY
HIGHER than normal.

So we embarked on the following project:


Unfortunately, stability isn't actually the largest invariance
condition that can be used for IME. N tail-related can be extended to
a more inclusive equivalence relation in stupidly ad hoc ways and
maintain IME usability.

HOWEVER, N tail-related and the N tail shift (both on Q^k) THEMSELVES
HAVE STRONG INVARIANCE PROPERTIES as subsets of Q^2k. So we look for

A) stability, N tail-related, N tail shift are the STRONGEST INVARIANT

This turns out to be only almost true. It can be completely fixed by
looking at simple natural strenghenings of IME, of the kind considered

August 16, 2018, #108, 66 pages.

In there, you find some elaborations on emulators. There are
duplicators, r-emulators, r-duplicators. If we use these for A) then
everything works as planned and A) becomes perfectly true. Of course,
A) is provably equivalent to Con(SRP) over WKL_0, and the forward
direction is provable in RCA_0.

To keep the reader well oriented, we repeat the TOC we have been using
up to now.


   2.1. Invariant Emulation Theory - #3 and here
      2.1.1. N, Z+, Q, Q[(a,b)], order equivalent, maximal emulator,
invariance - #3 and here
      2.1.2. N (sub)tail, N (sub)tail shift, N
tail-related - #3 and here
      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 - #6
      2.3.1. Positive, of S containedin Q[-n,n]^k, Q^k - #6
      2.3.2. Of S containedin Q[-n,n]^k, Q^k - #6

3. INDUCTIVE DOMAINS - in development
   3.1. Inductive Domains in Q
   3.2. Invariantly Inductive Domains in Q


My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
This is the 833rd 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

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

Harvey Friedman

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