# [FOM] 816: Beyond Perfectly Natural/15

Harvey Friedman hmflogic at gmail.com
Fri Jun 8 01:20:12 EDT 2018

```We now have the updated

106. Everybody’s Mathematics – Maximal Emulation Stability, 15
pages, June 7, 2018.

In [1], we have simplified the solitaire games considerably by
removing all of the "infinities" thee. We have also added FMES =
finite MES

EVERYBODY'S MATHEMATICS - MAXIMAL EMULATION STABILITY
by
Harvey M. Friedman
Distinguished University Professor of Mathematics,
Philosophy, Computer Science Emeritus
Ohio State University
Columbus, Ohio
June 7, 2018

Abstract. MES is Maximal Emulation Stability, the lead statement in
the new Emulation Theory. MES asserts that every finite subset of
Q[0,k]^k has a stable maximal emulator. MES is sufficiently natural,
transparent, concrete, elementary, interesting, memorable, teachable,
rich in varied intricate examples, and rich in weaker and stronger
forms, that it merits being classified in the category of Everybody's
Mathematics. MES is readily seen, via Gödel's Completeness Theorem, to
be implicitly Pi01. MES is independent of the usual ZFC axioms for
mathematics, being equivalent to Con(SRP), where SRP is ZFC together
with certain large cardinal hypotheses well investigated and accepted
by set theorists. We also discuss GMES = General Maximal Emulation
Stability, a natural transparent generalization, and Finite Maximal
Emulation Stability = FMES, both explicitly Pi01 and equivalent to
Con(SRP). We conclude with other explicitly Pi01 forms in the shape of
playable solitaire games. We argue that winning these games
constitutes confirmation of the consistency of mathematics.

1. Introduction.
2. Maximal Emulation Stability (MES).
3. Emulation Variants.
4. Stability Variants.
5. General MES.
6. Finite MES.
7. Finite Solitaire Games.
8. Confirming Consistency of Mathematics.
9. Formal Systems Used.
10. References.

************************************************************************
My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
This is the 816th 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
814: Beyond Perfectly Natural/13  6/3/18  2:05PM
815: Beyond Perfectly Natural/14  6/5/18  9:41PM

Harvey Friedman
```