[FOM] 805: Beyond Perfectly Natural/9

Harvey Friedman hmflogic at gmail.com
Fri Apr 20 22:47:11 EDT 2018

A new manuscript represent state of the art, with earlier versions
removed, has been put at


Here is the cover page:

Harvey M. Friedman
University Professor of Mathematics, Philosophy, Computer Science Emeritus
Ohio State University
Columbus, Ohio
April 20, 2018

ABSTRACT. We begin with the Upper Image Equation A U. R<[A] = [t]^k,
where R containedin [t]^k x [t]^k is the known and A containedin [t]^k
is the unknown. This Upper Image Equation (and obvious variants) is
easily seen to have a unique solution. Even for very explicit R (order
invariant), these unique solutions exhibit computational completeness.
Under a weakening of the equation (using "rich over"), we lose
uniqueness but gain a structure theory: for all order invariant R
containedin [kn]^k x [kn]^k, n >> k, there exists a nonempty stable A
containedin [kn]^k such that A U. R<[A] is rich over A. However, this
resulting Rich Stability statement is independent of the usual ZFC
axioms, and is, in fact, provably equivalent to Con(SRP) over EFA. RS
is explicitly P01 (using n >= (8k)!!).

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

My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
This is the 804th 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

Harvey Friedman

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