[FOM] 828: Tangible Incompleteness Restarted/2

[Harvey Friedman]

Here we continue with general perspectives before getting into the
mathematical details in FOM posting #829, Tangible Incompleteness
Restarted/3.

At this point we see three paradigm shifts that should result from the
successful full development of Tangible Incompleteness.

A. As a major expansion of the Goedel Incompleteness Phenomena. That
is, into simple basic transparent thematic discrete and finite
mathematical contexts which can be fully and readily engaged with even
mathematically gifted high school students. This is largely because
simple examples that require varied nontrivial mathematical arguments
of a basic elemental character are so readily generated. This is
indicative of the fundamental icharacter of the new Tangible
Incompleteness.

B. As a way of providing natural discrete and finite forms of large
cardinal hypotheses. See the discussion in section 2 below in this
posting.

C. As a way of blurring the commonly held distinction between
arithmetic sentences and set theoretic statements in terms of
objectivity and definite truth values. The most common view of
mathematicians outside mathematical logic is that there is a profound
definiteness to arithmetic statements such as Riemann hypothesis,
Goldbach's Conjecture, Kollatz Conjecture, twin prime conjecture, e+pi
is irrational, and so forth. They are true or false and there is no
two ways about it. Whereas for continuum hypothesis, Souslin's
hypothesis, there exists a measurable cardinal, every set of reals of
cardinality alpha_one is of measure zero, all real numbers are
constructible in the sense of Goedel, and so forth, there is no such
conviction. "It all depends on how you want to set up set theory".
Tangible Incompleteness can be viewed as BLURRING THE DISTINCTION
BETWEEN ARITHMETIC SENTENCES AND HIGHLY SET THEORETIC SENTENCES.

1. INVARIANT MAXIMALITY AT PRESENT

Invariance and Maximality SEPARATELY figure prominently throughout
mathematics as notions of clear intrinsic interest and power. The new
Invariant Maximality combines these notions in completely natural ways
with spectacular effect. Thus we ask for a maximal object (usually
obvious via Zorn's Lemma or direct construction) which ALSO satisfies
an invariance condition(s). The result is a developing coherent series
of tangible statements which can and can only be proved using
extensions of ZFC via large cardinal hypotheses. Although these
statements involve the existence of a countably infinite set
satisfying natural conditions, the statements are implicitly Pi01.
This can be seen from their logical form and the Goedel Completeness
Theorem. The foundational significance of being impolitely Pi01 is
that

a. It remains unprovable even if most of the great many well studied
celebrated set theoretic hypotheses (or their negations) are added to
ZFC such as V = L or conditions on the cardinality of the continuum,
Souslin's hypothesis, and so forth. This only requires implicit
arithmeticity. Also included in this would be regularity of projective
sets of reals - although here we do run into Con(inaccessible
cardinal) which however is too weak to effect our particular
implicitly arithmetic (even Pi01) statements.
b. It is demonstrably refutable in the sense that we know that if it
is false then it is provably false (in a very weak system).

A hallmark of physical theories is that generally speaking we know
that if they are false then they are refutable (by experimentation).
Physical theories that do not have this refutability property are
generally marginalized by the scientific community, and in particular
Nobel Prize committees.

It remains of great foundational importance to give fully natural
finite forms - i.e., where all objects under consideration are
finitary (although of course there are infinitely many finitary
objects under consideration). These finite forms appear in Part 3,
Sequential Constructions, as discussed above.

Presently, in Invariant Maximality, we work with the Q[-n,n]^k where
Q[-n,n] is the set of all rationals in [-n,n]. We treat 0,...,n as
preferred elements . We focus on subsets of Q[-n,n]^k. We use certain
families of subsets of Q[-n,n]^k. In most of our contexts it is
obvious by Zorn's Lemma or direct construction that these families of
subsets of Q[-n,n]^k have inclusion maximal elements.

So our lead statements in Invariant Maximality take the following form:

CERTAIN FAMILIES OF SUBSETS OF Q[-n,n]^k HAVE INCLUSION MAXIMAL
ELEMENTS WITH CERTAIN INVARIANCE.

We have found a particularly vivid notion of invariance that we use
for subsets of Q[-n,n]^k We define the N tail shift which maps each
Q^k into Q^k. We use the N tail shift invariance of S containedin
Q[-n,n]^k.

So here are the lead statements for Emulation Theory and Order
Invariant Graph Theory.

EMULATORS. Every finite E containedin Q[-n,n]^k has a completely N
tail shift invariant maximal emulator.

GRAPHS. Every order invariant graph G on Q[-n,n]^k has a completely N
tail shift invariant maximal clique..

So far the above development is quite close to our previous
developments. The main difference is the identification of, simple
definition of, and emphasis on, the very natural N tail shift. This is
the shifting by 1 (i.e., +1) of the N tail of the elements of Q^k.

There is also the closely related and crucially important equivalence
relation crit[k] on Q^k that we have used previously. This is not as
simple to define as the N tail shift. Nevertheless, we have made a
modest simplification in its definition using the N tail. We now call
this equivalence relation N tail similarity. So we also have these
versions:

EMULATORS. Every finite E containedin Q[-n,n]^k has an N tail
similarity invariant maximal emulator.

GRAPHS. Every order invariant graph G on Q[-n,n]^k has an N tail
similarity invariant maximal clique.

Since the N tail shift of x in Q^k is N tail similar to x, we see that
these last two lead statements respectively imply the previous two
lead statements.

These lead statements are easier to handle than previous lead
statements. They are stronger as the invariance notion is stronger.
Before I was not entirely happy with the simplicity of the invariance
conditions of this rough kind, and focused on looking for the weakest
possible invariance that could cause incompleteness. THAT IS STILL AN
IMPORTANT GOAL. However, at this point now that these stronger notions
of invariance have been fine tuned this well, it is best to use them.
This makes incompleteness easier to establish, but also has another
kind of advantage. IT NOW APPEARS THAT TINY DIMENSIONS ARE ENUF FOR
UNPROVABILITY OF ALL FOUR! CAUTION: This remains to be established.

In particular, we think that these stronger invariance conditions are
so strong that already for k = 2 we get Con(Z_2) and for k = 3 we get
far more than Con(ZFC). I think that very small n are needed for this.
Even maybe on Q[-2,2]^2 and Q[-3,3]^3, respectively. CAUTION: These
remain to be established.

For logically based classes of subsets of Q[-n,n]^k there are some new
twists and turns. For the stronger versions we naturally weaken the
notion of maximality. This is somewhat new territory.

Now once we have unprovability with N tail shift and N tail
similarity, with low dimension, under control, we can dig very deep to
see what tiny amount of invariance is really needed. This is where
rather massive complications surely arise.

2. INTERPRETING PRESENT INVARIANT MAXIMALITY AS:
DISCRETE FORMS OF LARGE CARDINAL HYPOTHESES

The analogs of complete N tail shift invariance and N tail similarity
invariance for ordinals in large cardinal theory are very clear and
very familiar in set theory. It corresponds to the subtle cardinal
hierarchy.

So our lead statements can be viewed as very natural implicitly Pi01
forms of large cardinal theorems. Of course the quantifiers in the
statements get all rearranged. But the skeleton is essentially the
same.

Let's take a look at one of them. The discussion is the same for the
other three.

GRAPHS. Every order invariant graph G on Q[-n,n]^k has an N tail
similarity invariant maximal clique.

What is the relevant large cardinal hypothesis?

LARGE CARDINAL. There exists a cardinal lambda such that for every B
containedin lambda^k there exists alpha_1 < ... alpha_n < lambda such
that B is {alpha_1,...,alpha_n} tail similarity invariant.

GRAPHS is naturally proved using LARGE CARDINAL, but with some ideas.
Not immediate by any means. Roughly speaking, lambda corresponds to
Q[-n,n], B corresponds to G, and alpha_1 < ... < alpha_n corresponds
to 1 < ... < n. We let lambda be a large cardinal. We weave the large
cardinal into a blown up version G* of G where we construct a maximal
clique S* in G* in a straightforward natural way (transfinite
construction of length lambda). We use this maximal clique S* to read
off B containedin lambda^k and apply LARGE CARDINAL to B. We get
alpha_1 < ... < alpha_n for the conclusion of LARGE CARDINAL. We then
extract an appropriate countable subsystem of
(G*,S*,alpha_1,...,alpha_n) which takes the form (G,S,1,...,n) where S
is a maximal clique in G, after applying an isomorphism that sends
each alpha_i to i.

We also give a finite form by giving a finite sequential constructions
associated with the four lead statements. Thus by transitivity we have
finite forms of large cardinal hypotheses.

These discrete and finite forms, as they proliferate and simplify,
should blur the distinction between arithmetic and set theoretic
statements in terms of their intrinsic objectivity (definite truth
values).

As we surmise in the next section, we think Invariant Maximality will
take on a life of its own, and not just be discrete forms of large
cardinal hypotheses.

3. INVARIANT MAXIMALITY IN THE FUTURE

There will be a vast subject called Invariant Maximality which cuts
across virtually the whole of mathematics. Weak forms of Invariant
Maximality, where the invariance is weakened, or where the dimension
is extremely low, and so forth, can be treated with various levels of
logical power well within ZFC, ranging from RCA_0 through ZC and
beyond. However, stronger forms even in low dimensions such as 3 or 4
cannot be handled in ZFC.

Invariant Maximality will become a major subdivision of mathematics,
rather singular in its connections with practically all established
mathematical contexts.

Virtually all mathematical contexts will be revisited with the lens of

a. classes of objects, at the core of the topic.
b. notions of refinement of the objects, at the core of the topic.
c. notions of maximally refined, at the core of the topic.
d. a catalog of invariance notions, at the core of the topic.
e. Determination of when the relevant classes of objects always have a
maximality refined element with the various invariance notions.
f. Use of high powered set theoretic methods (large cardinals) weakest
of which are available in ZFC, others not, for achieving e.

With some frequency f occurs. However even in contexts where high
powered set theoretic methods are completely avoided, there is still
much rich information, even decision problems, as to what invariance
conditions can be imposed.

Will large cardinals become an essential tool for proving theorems
that are not ostensibly connected with Emulators, Maximal Cliques,
Logical Classes, and so forth?

We believe that the answer to this is yes, and will occur in stages.
At some point this century, large cardinals via Tangible
Incompleteness will be used to give the first proof of some
conjectures far removed from Emulators, Maximal Cliques, Logical
Classes, etc. They will later be removed. This has already happened in
a different way with work of Richard Laver in connection with left
distributive algebras. However, at some point such applications of
large cardinals via Tangible Incompleteness will be unremovable.

History of mathematics tends to show that when a sufficiently
fundamental and attractive new kind of mathematics is developed, it
eventually finds its way to unexpected applications. Tangible
Incompleteness is itself such an application, but it begs to be woven
into the very fabric of general mathematical activity.

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This is the 828th 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=CNEQ_ftUXkk8vw06ur2ZIz-fKtgBlRWVnt2FURhDviQ&s=lTkr9hQsCiXHPdA3eKvor1lzSvejoyj72ocJz-skUOo&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
827: Tangible Incompleteness Restarted/1

Harvey Friedman