# [FOM] 814: Beyond Perfectly Natural/13

Harvey Friedman hmflogic at gmail.com
Sun Jun 3 14:05:58 EDT 2018

```In https://cs.nyu.edu/pipermail/fom/2018-May/021019.html I discussed
my recent talk at
#69.

At the end of that posting, I discussed a finite form of MES that
appears at the end of that talk, page 32. On page 10 of the talk I
warn everybody that I don't want to talk about 12 hour old "results".
It turns out that that was good advice. I have struggled quite a bit
to find better and better finite forms in Emulation Theory, with
plenty of false starts. So I retract page 32 (as indicated on my
website), and also the corresponding last part of the previous FOM
posting #813.

Here is a new version, full of confidence, which I LIKE BETTER ANYWAY.

*FINITE MES*
in {0,...,kn}^k

Let E containedin {0,...,kn}^k. S is an emulator of E if and only if S
containedin {0,...,kn}^k and every element of S^2 is order equivalent
to an element of E^2. S is a maximal emulator of E containedin
{0,...,kn}^k if and only if S is an emulator of E containedin
{0,...,kn}^k which is not a proper subset of any emulator of E
contaiendin {0,...,kn}^k. We can say this more vividly this way: S is
ruined by any new element from {0,...,kn}^k.

E is stable if and only if for all 0 <= i < n, (i,n,2n,...,(k-1)n) in
E if and only if (i,2n,3n,...,kn) in E.

The mix of E is the set of all elements of {0,...,kn}^k whose
coordinates are all coordinates of elements of E.

FINITE MAXIMAL EMULATION STABILITY. FMES. Assume n > (8kr)!. Every
subset of {0,...,kn}^k has a length r tower of stable emulations,
where each term is ruined by any new element from {0,n,2n,...,kn}^k
and by any new element from the mixes of the previous terms.

FMES is obviously explicitly Pi01.

THEOREM 1. FMES, FMES for r = 3 are both equivalent to Con(SRP) over
EFA. There is a hierarchy through Con(SRP) obtained by fixing k.

************************************************************************
My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
This is the 814th 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  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

Harvey Friedman
```