[FOM] 813: Beyond Perfectly Natural/12

[Harvey Friedman]

On 5/29/18 I give a talk at a local CS meeting. The pdf displayed for
the talk is going to be at
https://u.osu.edu/friedman.8/foundational-adventures/downloadable-lecture-notes-2/
This Foundationalist Looks at P = NP, #69

Also this week I am placing a new ms. at
https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
called INCOMPLETENESS IN EVERYBODY'S MATHEMATICS.

Recall MES:

EVERY FINITE SUBSET OF Q[0,k]^k HAS A STABLE MAXIMAL EMULATOR.

The idea is that the newish MAXIMAL EMULATION STABILITY = MES is NOW
so good that it warrants to be viewed as part of "everybody's
mathematics" even though it is new. SMALL CHANGES in terminology,
emphases, comments, exposition, presentation, etcetera, have made and
are continuing to make a huge difference in this fairly advanced stage
of the 50 year Concrete Mathematical Incompleteness program.

In the talk I give, I am committing myself to getting serious
engagement on MES with GIFTED HIGH SCHOOL students, in 2019-2020. The
MES statement is so basic that no significant knowledge is required to
fully wrap one's head around it and see it clearly. If this proves
successful in the way I am planning it, we would have the following
result:

THEOREM. There exists an implicitly Pi01 sentence with the following properties.
1. It is equivalent to the consistency of SRP over WKL_0.
2. A significant number of Gifted High School students have
meaningfully engaged in the sentence, as evidenced by 3,4 below.
3. Successful Completion of relevant homework exercises demonstrating
basic competence in the underlying notions and basic facts.
4. Such exercises include the building of maximal emulators, and
checking them for stability.

In preparing this talk, my current thinking is to try to squeeze out
as much as possible from the MAXIMAL EMULATION setup. So I have been
THINKING about giving an EXPLICITLY Pi01 statement in this
environment. This appears to be rather successful:

DEFINITION 1. S is an emulator of E containedin {0,...,kn}^k if and
only if S containedin {0,...,kn}^k and every element of S^2 is order
equivalent to an element of E^2.

DEFINITION 2. S is a weakly maximal emulator of E
containedin {0,...,kn}^k if and only if S is an emulator of E
containedin {0,...,kn}^k such that for all emulators S U {x,ny} of E
containedin {0,...,kn}^k, (x,ny) is order equivalent to some (x',ny)
in S^2.

DEFINITION 2. S containedin {0,...,kn}k is stable if and only if for
all 0 <= p < n, (p,n,2n,...,(k-1)n) in S if and only if
(p,2n,3n,...,kn) in S.

FINITE MAXIMAL EMULATION STABILITY. FMES. Assume n > (8k)!. Every
subset of {0,...,kn}^k has a stable weakly maximal emulator.

THEOREM. FMES is provably equivalent to Con(SRP) over EFA.

So why not sit forever in this MAXIMAL EMULATION ENVIRONMENT?

Well, at the moment the prospect of getting to HUGE in this
environment seems daunting, and I don't want to take the plunge. At
least not until I write up the present way of getting to HUGE which is
fairly but not *hugely* (pardon the pun) successful. (But I am tempted
to take the plunge, as this environment is looking to be surprisingly
entertaining).

************************************************************************
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 813th 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
http://u.osu.edu/friedman.8/foundational-adventures/fom-email-list/

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

Harvey Friedman