[FOM] 745: Philosophy of Incompleteness/7

[Harvey Friedman]

So we now have the ingredients for how we want to proceed in light of
the very poor prospects for our

A. An existing mathematical question that is widely known and of wide
interest, is shown to be neither provable nor refutable in ZFC.

We of course want to move to the earlier stated

SITUATION TO BE ENVISIONED. The typical mathematically literate
INTELLECTUAL, but not professional mathematician, becomes at least as
or even more interested in some new AREA of concrete mathematics than
they are in the typical so called breakthrough result in the various
standard special areas of mathematics, AND that major results in that
AREA can only be established by going way beyond ZFC. And INTELLECTUAL
says to the mathematicians that "the AREA involves objects even
simpler and more basic than what you typically consider, and the
statements in the AREA are even clearer and more well motivated". So
the fact that YOU, mathematician, ignore this new AREA because you
didn't create it, and wish to avoid foundational issues at all costs,
is not convincing to the INTELLECTUAL. In fact, the INTELLECTUAL sees
an inevitable path to the new AREA that cannot be rationally resisted,
at least in the long run.

As the avid FOM reader knows, my best shot towards the SITUATION TO BE
ENVISIONED is thorough EMULATION THEORY.

Now there is definitely already or going to be a small group of people
either pretty steeped in f.o.m. or maybe having a lot of personal
contact with me and having some familiarity with f.o.m., who are
already pretty convinced that Emulation Theory has much of the
trappings of the SITUATION. Here we use SITUATION to abbreviation the
above SITUATION TO BE ENVISIONED.

Of course, this is far from what we really want to have. But in my
opinion the basic ingredients for the SITUATION are already present in
Emulation Theory. The ideas just need to be refined and developed
further in certain ways to make it yet more convincing.

We could look for a single statement independent of ZFC that does not
really constitute any kind of new Area of Research. However, the
standards for this having the desired force are incredibly high, as it
can so easily be dismissed as a one off curiosity that can so safely
be ignored. Of course, it can still be so spectacularly overwhelming
that it still has the desired force. But the prospects for such a
thing are poor, at least compared to the creation of an appropriate
new Area of Research.

Now that we are committed to creating an appropriate new Area of
Research, there are two main strategies, which of course can be
combined.

1. Create a short path from existing respected mathematics to the new
Area. I.e., start with some already celebrated theorem or group of
theorems, and modify them or relate them in some way to the new Area.
This may involve first restating the existing mathematics in a
different form or with different terminology, where this would be
generally regarded as a purely expositional restatement - obviously
with an unexpected purpose in mind (as a first step to relating to the
new Area). Then the crucial injection of another ingredient takes
place. The added ingredient should also stand on its own as
fundamentally simple and ideally appearing as an ingredient across the
mathematical landscape.

2. Create a motivating thematic metaphor relating the Area to a
familiar situation which makes clear familiar sense and is fundamental
outside mathematics.

Emulation Theory follows 1,2 to some real extent, and we will be doing
some more work to strengthen 1,2.

We will concentrate first on strategy 1.

SHORT PATH TO EMULATION THEORY

For most mathematicians and others, the least familiar ingredient of
Emulation Theory is DROP EQUIVALENCE. This is the conclusion or punch
line in most Emulation Theory statements. This is a kind of symmetry.

We start by relating Drop Equivalence to already celebrated
combinatorial mathematics. The link is sufficiently strong that, in
fact, Drop Equivalence is already in essence familiar to at least some
combinatorists and many logicians. But in any case, it is an extremely
simple and natural idea.

In order to make Drop Equivalence most vivid, we will adhere to
dimension 2 only. It will then be obvious how to lift it to any
dimension k >= 2.

Let S be a subset of N^2, where N is the set of all nonnegative
integers. Let (n,r) be in N^3. We can drop from (n,r) down to the x
axis, where we start with (n,r-1), and continuing with
(n,r-2),...,(n,0), and then we stop. Now, generally speaking, some of
these pairs will lie in S whereas others will not. So as we drop
through these r points on the way down, we will get an in/out pattern
in S.

Now suppose we have another pari (m,r), with the same second
coordinate r. We can compare the in/out pattern of these two:

(n,r-1),(n,r-2),...,(n,0).
(n,r-1),(n,r-2),...,(n,0).

If we get the same in/out pattern (in and out of S), then we say that
S is drop equivalent at (n,r),(m,r). Of course, drop equivalence for a
given S is indeed an equivalence relation both for fixed r and varying
r.

Now obviously no matter what S is, we automatically have the trivial
drop equivalence at all (n,0),(m,0) since 0 is least.

THEOREM 1.1. Let S containedin N^3. For all r, S is drop equivalent at
some two distinct (n,r),(m,r).

Proof: By an obvious pigeonhole argument (very celebrated, natural,
well known). QED

So far, this is a particular friendly arrangement of a very well
known, celebrated situation.

Now let's get more aggressive and use (n,r),(r,r). We run into
trouble. In fact, we run into trouble no matter what r > 0 we use.

THEOREM 1.2. The following is false. Every S containedin N^2 is drop
equivalent at some distinct (n,r),(r,r), r > 0.

Proof: Let S = {(t+1,t): t in N}. Then for r > 0, we have exactly one
r such that (r,r-1) in S. The obvious problem is that S can talk about
r-1. QED

It is now clear that the problem with getting nontrivial drop
equivalence at (n,r),(r,r) is the lack of limits points in (N,<).

So we need to bring limit points into the picture. We make the more
general definition of drop equivalence.

THEOREM 1.3. The following are false. Let J be an interval in Q and J'
be an interval in R.
i. Every S containedin J^2 is drop equivalent at some distinct
(x,y),(y,y), y not least.
ii. Every S containedin J'^2 is drop equivalent at some distinct
(x,y),(y,y), y not least.

Set theorists have a very celebrated and well known (to set theorists)
way of getting around this.

THEOREM 1.4. We can go well beyond ZFC in well known and well studied
ways (by set theorists) to get a linear ordering (D,<) where every S
containedin D^2 is drop equivalent at some distinct (x,y),(y,y), y not
least. Over ZFC, we can prove that any such D has cardinality that of
a strongly inaccessible cardinal and much more. In fact, we can use
the first subtle cardinal with the usual <, and that is the least
cardinality of any such (D,<). We can also use a dense linear ordering
with endpoints.

Of course, this (D,<) is way out there, unimaginably enormous, not
within the scope of ZFC.

HOWEVER, is there a version of such a (D,<) that is familiar and down to earth?

Of course ideally, we would like to use, say, Q[0,1] which is the unit
interval of rational numbers with endpoints 0,1. By Theorem 1.3, we
cannot literally use such a concrete linear ordering.

But can we get down to the combinatorial essence of what is going on
in Theorem 1.4? A clue that there may be an opportunity to do just
that is afforded by the realization that we have been looking for
(D,<) where ALL S containedin D^2 has the requisite property. The
obvious approach would be to weaken the "all S" here.

In the next posting we will shos how to bring this down to earth.

************************************************************************
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 745th 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 #1-699 can be found at
http://u.osu.edu/friedman.8/foundational-adventures/fom-email-list/

700: Large Cardinals and Continuations/14  8/1/16  11:01AM
701: Extending Functions/1  8/10/16  10:02AM
702: Large Cardinals and Continuations/15  8/22/16  9:22PM
703: Large Cardinals and Continuations/16  8/26/16  12:03AM
704: Large Cardinals and Continuations/17  8/31/16  12:55AM
705: Large Cardinals and Continuations/18  8/31/16  11:47PM
706: Second Incompleteness/1  7/5/16  2:03AM
707: Second Incompleteness/2  9/8/16  3:37PM
708: Second Incompleteness/3  9/11/16  10:33PM
709: Large Cardinals and Continuations/19  9/13/16 4:17AM
710: Large Cardinals and Continuations/20  9/14/16  1:27AM
700: Large Cardinals and Continuations/14  8/1/16  11:01AM
701: Extending Functions/1  8/10/16  10:02AM
702: Large Cardinals and Continuations/15  8/22/16  9:22PM
703: Large Cardinals and Continuations/16  8/26/16  12:03AM
704: Large Cardinals and Continuations/17  8/31/16  12:55AM
705: Large Cardinals and Continuations/18  8/31/16  11:47PM
706: Second Incompleteness/1  7/5/16  2:03AM
707: Second Incompleteness/2  9/8/16  3:37PM
708: Second Incompleteness/3  9/11/16  10:33PM
709: Large Cardinals and Continuations/19  9/13/16 4:17AM
710: Large Cardinals and Continuations/20  9/14/16  1:27AM
711: Large Cardinals and Continuations/21  9/18/16 10:42AM
712: PA Incompleteness/1  9/2316  1:20AM
713: Foundations of Geometry/1  9/24/16  2:09PM
714: Foundations of Geometry/2  9/25/16  10:26PM
715: Foundations of Geometry/3  9/27/16  1:08AM
716: Foundations of Geometry/4  9/27/16  10:25PM
717: Foundations of Geometry/5  9/30/16  12:16AM
718: Foundations of Geometry/6  101/16  12:19PM
719: Large Cardinals and Emulations/22
720: Foundations of Geometry/7  10/2/16  1:59PM
721: Large Cardinals and Emulations//23  10/4/16  2:35AM
722: Large Cardinals and Emulations/24  10/616  1:59AM
723: Philosophical Geometry/8  10/816  1:47AM
724: Philosophical Geometry/9  10/10/16  9:36AM
725: Philosophical Geometry/10  10/14/16  10:16PM
726: Philosophical Geometry/11  Oct 17 16:04:26 EDT 2016
727: Large Cardinals and Emulations/25  10/20/16  1:37PM
728: Philosophical Geometry/12  10/24/16  3:35PM
729: Consistency of Mathematics/1  10/25/16  1:25PM
730: Consistency of Mathematics/2  11/17/16  9:50PM
731: Large Cardinals and Emulations/26  11/21/16  5:40PM
732: Large Cardinals and Emulations/27  11/28/16  1:31AM
733: Large Cardinals and Emulations/28  12/6/16  1AM
734: Large Cardinals and Emulations/29  12/8/16  2:53PM
735: Philosophical Geometry/13  12/19/16  4:24PM
736: Philosophical Geometry/14  12/20/16  12:43PM
737: Philosophical Geometry/15  12/22/16  3:24PM
738: Philosophical Geometry/16  12/27/16  6:54PM
739: Philosophical Geometry/17  1/2/17  11:50PM
740: Philosophy of Incompleteness/2  1/7/16  8:33AM
741: Philosophy of Incompleteness/3  1/7/16  1:18PM
742: Philosophy of Incompleteness/4  1/8/16 3:45AM
743: Philosophy of Incompleteness/5  1/9/16  2:32PM
744: Philosophy of Incompleteness/6  1/10/16  1/10/16  12:15AM

Harvey Friedman