# [FOM] 806: Beyond Perfectly Natural/10

Harvey Friedman hmflogic at gmail.com
Sun Apr 22 21:06:40 EDT 2018

```There is a new level of much higher thematic and conceptual clarity
and naturalness in the new ms.

103 Upper Image Stability, 7 pages. April 22, 2018. Greatly improved
thematically and conceptually from 102, 4/20/18, though 102 is worth
keeping.

Here is the cover page:

UPPER IMAGE STABILITY
by
Harvey M. Friedman
Distinguished University Professor of Mathematics, Philosophy,
Computer Science Emeritus
Ohio State University
Columbus, Ohio
April 22, 2018

ABSTRACT. We begin with the Upper Image Equation R<[A] = A', where R
containedin [t]^k x [t]^k is the known and A containedin [t]^k is the
unknown. This Upper Image Equation is easily seen to have a unique
solution. For order invariant R, these unique solutions exhibit
computational completeness. We seek "stable" A containedin [kn]^k, n
>> k, in the sense that for all p in [n-1], (p,n,2n,...,kn-n) in A iff
(p,2n,3n,...,kn) in A. This requires that we use an approximate form
of the Upper Image Equation R<[A] = A'. Upper Image Stability is
proved only by going well beyond the usual ZFC axioms for mathematics.
In fact, Upper Image Stability is provably equivalent to Con(SRP) over
EFA, and is explicitly Pi01 (using n >= (8k)!!).

1. Definitions.
2. Upper Image Equation.
3. Upper Image Stability.
4. Proof Sketch.

This FEELS like a watershed moment where the Explicitly Pi01 has
finally stabilized (not meant as a pun).

I wonder if readers can really feel the differences. And if not, how
that difference can be explained. Even the difference between 102
(still there) and this 103 is a sea change event.

I am increasingly captivated by the analogy with musical composition.
At various stages you have perfectly fine playable material, but you
incessantly look for more, and feel like you don't quite have the
exactly right notes. Then it all gels and you are led to the obviously
greatly improved material - obsoleting so much earlier material.

Continuous 50 year mathematical composition efforts are obviously very
rare, In music one is familiar with this, but of course for long but
not this long - time periods. There is a very famous and excellent
video of Leonard Bernstein concerning the "right notes" and the
"principle of inevitability":
lectures on "right notes/inevitability" and they are inspiring.

WHAT REMAINS?

1. For Explicitly Pi01. We see the forward direction. For the
reversal, I have a technique which creates material for use in the
compactness theorem,  by a sequential iteration argument. After
compactness is applied, one arrives with an INFINITE form of the Upper
Image Stability statement. Then the techniques from other parts of the
CMI project (concrete mathematical incompleteness) kick in, related to
upper shifting.

2. For Explicitly Pi01. What about explicitly Pi01 corresponding to
HUGE? Now that the standards for explicitly Pi01 have gone way up,
this seems like a daunting task.

3. For Implicitly Pi01. This has proved to be easier. The main ones
are on the rational interval Q[0,k].

In fact, there has been a kind of merging of 3 with the present. Here
is the MAIN EMULATION THEORY statement in new terminology.

DEFINITION. Let A containedin Q[0,k]^k. A is stable if and only if for
all p < 1, (p,1,...,k-1) in A iff (p,2,...,k) in A.

MAXIMAL EMULATION STABILITY. MES. Every finite subset of Q[0,k] has a
stable maximal emulation.

THEOREM. MES 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
This is the 806th 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