# [FOM] 834: Tangible Incompleteness Restarted/8

Harvey Friedman hmflogic at gmail.com
Thu Oct 10 17:02:05 EDT 2019

```We didn't put

[1] 108. Tangible Incompleteness, Interim Report, October 6, 2019.

up yet on my website. In preparing this manuscript we had some major
new insights and simplifications, and want to give the highlights now
before I finish [1].

Recall that the crucial issue I have been struggling with ever since I
discovered Emulation Theory is with

IME. Invariant Maximal Emulation. Every finite subset of Q[-n,n]^k has
an INVARIANT maximal emulator.

What is the simplest natural Invariance here that resonates with
practically anyone and that also can be justified abstractly among all
alternatives of a certain kind?

Let x in Q^k. A fractional coordinate of x is a coordinate that is not
an integer.

1. THE SHIFT. Add 1 to all coordinates of x.
2. THE N SHIFT. Add 1 to all nonnegative integer coordinates of x.
3. THE N TAIL SHIFT. Add 1 to all nonnegative integer coordinates of x
that are greater than all fractional coordinates of x.

IME (bad). Invariant Maximal Emulation (bad). Every finite subset of
Q[-n,n]^k has a completely shift invariant maximal emulator.

IME (bad). Invariant Maximal Emulation (bad). Every finite subset of
Q[-n,n]^k has a completely N shift invariant maximal emulator.

IME. Invariant Maximal Emulation. Every finite subset of Q[-n,n]^k has
a completely N tail shift invariant maximal emulator.

THEOREM. (RCA_0) The two bad IME's are refutable in RCA_0. IME is
provably equivalent to Con(SRP) over WKL_0. The forward direction is
provable in RCA_0.

In [1] we discuss the special status of the N tail shift in Emulation
Theory. We will stop here and point the reader to [1] which should be
available October 11, 2019.

************************************************************************
My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
This is the 833rd 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
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
822: Tangible Incompleteness/1  7/14/18  10:56PM
823: Tangible Incompleteness/2  7/17/18  10:54PM
824: Tangible Incompleteness/3  7/18/18  11:13PM
825: Tangible Incompleteness/4  7/20/18  12:37AM
826: Tangible Incompleteness/5  7/26/18  11:37PM
827: Tangible Incompleteness Restarted/1  9/23/19  11:19PM
828: Tangible Incompleteness Restarted/2  9/23/19  11:19PM
829: Tangible Incompleteness Restarted/3  9/23/19  11:20PM
830: Tangible Incompleteness Restarted/4  9/26/19  1:17 PM
831: Tangible Incompleteness Restarted/5  9/29/19  2:54AM
832: Tangible Incompleteness Restarted/6  10/2/19  1:15PM
833: Tangible Incompleteness Restarted/7  10/5/19  2:34PM

Harvey Friedman
```