[FOM] 822: Tangible Incompleteness/1

[Harvey Friedman]

NEW MANUSCRIPT AT
http://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
   #106  July 14b, 2018

I have been busy trying to put the finishing touches on the lead event
in Emulation Theory. Here is the lead event:

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

Things have finally stabilized (pun intended) with what appears to be
just the right notion of STABILITY. Strong enough to make MES (and
also MDS and MCS) independent of ZFC (actually equivalent to Con(SRP)
over WKL_0). Simple enough to stay in EVERYBODY'S MATHEMATICS.

OH!!, what happened is that my ORIGINAL notion of stable was

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

turns out to be insufficient for my reversal techniques to show
independence of ZFC, as I have claimed. So it needs to be strengthened
WITHOUT RUINING EVERYBODY'S MATHEMATICS.

NEW STABILITY

2) S containedin Q[0,k]^k is stable if and only if for all p_1,...,p_i
< i, (p_1,...,p_i,i,...,k-1) in S iff (p_1,...,p_i,i+1,...,k) in S.

REMARKS

Obviously 1) is the same as 2) with just i = 1.

2) turns out to be far more than what is really needed. I will be
reversing ultimately with far less. For example, one target is to
reverse with 2) only for i = 1,2.

2) is very explicit, with all coordinates nicely displayed, with nice
consecutively, etc. But there is also a very nice strengthening, which
we call stableUP. Here UP is an up arrow.

STABLEUP

3) S containedin Q[0,k]^k is stableUP if and only if no matter how you
extend any given length 0 <= i <= k sequence of rationals by k-i
strictly greater strictly increasing integers from {1,...,k},
membership in S is equivalent.

3) is readily grasped, period, or at least after one has absorbed 2).

BUT WHAT is the ultimate form of this kind of stability that works
with MES? It is called FULLY STABLE.

I have been engaged in two STRUGGLES concerning fully stable.

a. What is the most transparent way of defining fully stable? You will
find fully stable in a little different context of Q[0,1]^k, involving
some less than optimal terminology, treated in my Putnam volume paper
http://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
#92 I.e., MES, using it, is shown to follow from Con(SRP) there.

b. Maybe we can characterize the "basic" equivalence relations that
can be used in MES? Of course, such a result will require large
cardinals.

PREFERRED DEFINITION OF FULLY STABLE

DEFINITION. x,y in Q^k are critically equivalent if and only if x,y
are order equivalent, and their strictly increasing enumerations both
lie in {1,...,k} at or past any position at which they differ.

S containedin Q[0,k]^k is strongly stable if and only if for all
critically equivalent x,y in Q[0,k]^k, x in S iff y in S.

SUITABLY DEFINABLE EQUIVALENCE RELATIONS

We expect that a complete characterization of the equivalence
relations on Q^k that are quantifier free definable over (Q,Z+,<) and
can be used in MES, is feasible. Of course, this is going to require
large cardinals.

PRESENT TOP PRIORITY

Everything is on hold, including:

1. Finite to Infinite.
2. Strict Reverse Mathematics.
3. Reworking Goedel's Second Incompleteness.
4. Sugared ZFC.
5. Several others...

until I finish a solid manuscript on the reversal of MES with fully
stable. And then immediate thereafter, with stable in form 2).

After that, I can breathe a little and move on 1-5 AND this
characterization of equivalence relations project.

************************************************************************
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 822nd 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
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
816: Beyond Perfectly Natural/15  6/8/18  1:20AM
817: Beyond Perfectly Natural/16  Jun 13 01:08:40
818: Beyond Perfectly Natural/17  6/13/18  4:16PM
819: Sugared ZFC Formalization/1  6/13/18  6:42PM
820: Sugared ZFC Formalization/2  6/14/18  6:45PM
821: Beyond Perfectly Natural/18  6/17/18  1:11AM

Harvey Friedman