[FOM] 827: Tangible Incompleteness Restarted/1

Mon Sep 23 23:19:12 EDT 2019

I now come with a refined perspective on Tangible Incompleteness after
a year's hiatus (recently focusing on my youtube site). Breakthroughs
also are starting to come in.

The prospective book is to be tentatively called TANGIBLE
INCOMPLETENESS. Tentatively, it will be divided into 3 parts.

PART 1. BOOLEAN RELATION THEORY
PART 2. INVARIANT MAXIMALITY
PART 3. SEQUENTIAL CONStRUCTIONS.

Part 1 is already in polished form on my webpage and is quite
substantial - about 800 pages of material. Parts 2,3 remain to be
written.

Invariant Maximality includes the Emulation Theory that we have been
discussing in the previous postings. We have found two other theories
that are cousins of Emulation Theory and the common thread is that of
Invariant Maximality. Thus

PART 2. INVARIANT MAXIMALITY
   2.1. EMULATION THEORY
   2.2. INVARIANT GRAPH THEORY
   2.3. LOGICAL INVARIANT THEORY

2.1 and 2.2 are closely related, and we establish an equivalence
between the two. Emulation theory with its maximal emulations, has
perhaps the broadest potential appeal among minimally mathematical
thinkers, the general mathematician, and the gifted high school
student. This is because the outermost universal quantifier is over
the very basic finite sets of rational vectors of fixed dimension, and
the notion of emulator can be described informally and thematically
without getting into the details (and the details are transparent). In
fact we believe that emulation theory in dimension 3 already goes far
beyond ZFC, as is invariant graph theory and logical invariant theory
- although this has yet to be established. However, only emulation
theory has such a trivial outermost quantifier. But all three are
readily understood by those comfortable with undergraduate level
mathematics.

In emulation theory, practically any small set E of rational vectors
in low dimension leads to a very interesting rich development. S
emulates E if and only if any two elements of S "look alike" some two
elements of E. We look for inclusion maximal emulators with certain
invariance properties. This should support a very effective and
exciting curriculum at the mathematically gifted high school level,
and the plan is to develop this for use in the main Summer School
programs for students at that level.

For related reasons, emulation theory may be key for realizing the
future of Invariant Maximality that I discuss below. Instead of just
working with (finite) sets of rational vectors, we can work with
(finite) subsets of the domain of a relational structure. There is a
clear notion of emulator here.

However, for many discrete mathematicians, especially graph theorists,
Invariant Graph Theory may look more attractive. Instead of emulators
of finite sets, we look at cliques in (order invariant) graphs. We
look for maximal cliques with certain invariance properties. But I
have found a somewhat surprising reaction against graphs and cliques
among many mathematicians. Notable is the statement made to me by a
Fields Medalist when asked if they knew what a graph is. ANSWER: No I
don't, and I never want to know what a graph is. So the interesting
question here is just what did this Fields Medalist know about graphs
to make them never want to know what a graph is? Let me repeat for
emphasis:

WHAT DID THIS FIELDS MEDALIST KNOW ABOUT GRAPHS TO MAKE THEM NEVER
WANT TO KNOW WHAT A GRAPH IS?

The answer to this question should reveal a lot about the current
mathematical environment.

We would expect that for logicians, especially model theorists,
Logical Invariant Theory would resonate the most. Here we use
universal classes, or A...A classes of structures, and also A...AE...E
classes of structures. We ask for an element which is maximal in
various senses, with certain invariance properties. Many logicians,
particularly many of the model theorists, prefer universal classes.
Computer scientists might be split between universal classes and
maximal cliques. General mathematicians and non mathematicians and
gifted students would prefer maximal emulations. I expect that the
further away from math the greater the affinity for emulators.

Of course the dramatic point is that the basic results of Emulation,
Invariant Graph, and Logical Invariant theory are unprovable in ZFC
and in fact are provably equivalent to the consistency of certain
large cardinal hypotheses, most commonly Con(SRP). They are implicitly
Pi01 due to their logical form, via the Goedel Completeness Theorem.
We believe that the phenomena kicks in already in dimension 3,
although this is yet to be established.

Thus the reach of the Incompleteness Phenomena has been greatly
extended by such totally clear and accessible thematic examples of
Tangible Incompleteness.

However there is an additional way of looking at these results. Namely as

DISCRETE FORMS OF LARGE CARDINAL HYPOTHESES.

This point is discussed in section 2 of FOM posting #828, Tangible
Incompleteness Restarted/2.

We now come to Part 3, Sequential Constructions. Every one of our main
statements lead rather naturally to a simple sequential construction.
The main statement is easily seen to be equivalent to asserting that
this simple sequential construction can be carried out for infinitely
many steps - as an infinitely long sequential construction. The
challenge is to transparently give appropriate finite approximations.
If we simply say that the construction can be carried out for any
given finite number of steps then we get a statement which is too
weak, even provable in weak fragments of PA. However, if we also
demand that we maintain "balance", a very natural requirement on the
rationals being used, then we obtain an explicitly Pi01 sentence
equivalent to Con(SRP).

We have found that we can obtain great leverage by allowing such
sequential construction to be a bit more technical. Still nowhere near
as technical as what is routinely done in combinatorics and algorithm
courses even at the undergraduate level. And with this great leverage
comes the full large cardinal hierarchy, with explicitly Pi01
sentences asserting that certain constructions can be carried out for
any finite number of steps!

Of course the plan is also to keep the conditions used in these finite
sequential constructions so simple that they are subject to a calculus
of conditions and decision procedures. I.e., determination of which
simple conditions for the sequential constructions allow the
sequential construction to be carried out for any finite number of
steps.

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My website is at https://urldefense.proofpoint.com/v2/url?u=https-3A__u.osu.edu_friedman.8_&d=DwIBaQ&c=slrrB7dE8n7gBJbeO0g-IQ&r=xXZM6ZrkjVxXknjzIxhAvQ&m=3X4dk9jn2VcDD7bIXcMAoMUT0-6Nn9dHvHv0uR2NEB4&s=lTzd6TFmW08nTSrbEcHFG7lXxsT4JFqDvB47WlcBrVM&e=  and my youtube site is at
https://urldefense.proofpoint.com/v2/url?u=https-3A__www.youtube.com_channel_UCdRdeExwKiWndBl4YOxBTEQ&d=DwIBaQ&c=slrrB7dE8n7gBJbeO0g-IQ&r=xXZM6ZrkjVxXknjzIxhAvQ&m=3X4dk9jn2VcDD7bIXcMAoMUT0-6Nn9dHvHv0uR2NEB4&s=ZwaDiBF3B2tAfOfbJLzHdqmqKALBltxZKuQ9ZcScajQ&e= 
This is the 827th 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
https://urldefense.proofpoint.com/v2/url?u=http-3A__u.osu.edu_friedman.8_foundational-2Dadventures_fom-2Demail-2Dlist_&d=DwIBaQ&c=slrrB7dE8n7gBJbeO0g-IQ&r=xXZM6ZrkjVxXknjzIxhAvQ&m=3X4dk9jn2VcDD7bIXcMAoMUT0-6Nn9dHvHv0uR2NEB4&s=kbWhJmf5-EqWzfqd6G_PAzda2colGhpdA-XoJ5GqcPk&e= 

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

Harvey Friedman