[FOM] 791: Emulation Theory/Inductive Equations/1

Harvey Friedman hmflogic at gmail.com
Thu Feb 8 23:52:32 EST 2018

```We restart the part of
https://cs.nyu.edu/pipermail/fom/2017-December/020746.html concerning
Concrete Mathematical Incompleteness. The paper

#92

is to appear, and will not be modified to take into account major
subsequent developments.

We now have several breakthroughs beyond
https://cs.nyu.edu/pipermail/fom/2017-December/020746.html

Here is the front end of the paper

Concrete Mathematical Incompleteness Status 2/8/18
#98

CONCRETE MATHEMATICAL INCOMPLETENESS STATUS 2/8/18
by
Harvey M. Friedman
University Professor of Mathematics, Philosophy, Computer Science Emeritus
Ohio State University
Columbus, Ohio
February 8, 2018

The book is expected to have three major parts:

CONCRETE MATHEMAITCAL INCOMPLETENESS

PART 1. BOOLEAN RELATION THEORY.
PART 2. EMULATION THEORY.
PART 3. INDUCTIVE EQUATION THEORY.

1. Emulation Theory/infinite/SRP.
2. Inductive Equation Theory/infinite/SRP.
3. Inductive Equation Theory/infinite/HUGE.
4. Inductive Equation Theory/finite/SRP.
5. Inductive Equation Theory/finite/HUGE.

I have recently been determined to move to functions rather than stick
with sets. However, I have now realized how I can do more with sets
than I had previously thought, and certain things are more naturally
transparent with sets.

So I am now moving back to sets throughout.

######

EXTENDED REMARKS

In this posting, I want to make some extended remarks.

I view these as the advanced stages after about 50 years of progress: :

...
...
...

STAGE n. Perfectly Natural, Simple, Transparent, Easily Explained,
etcetera. Always implicitly Pi01, and sometimes explicitly Pi01, and
always independent of ZFC.
STAGE n+1. Of clear interest to a significant portion of the
mathematical community, as pieces of mathematics, independent of the
significance for the foundations of mathematics. Always implicitly
Pi01, and sometimes explicitly Pi01, and always independent of ZFC.
Measured how? By seminar talks
and colloquium lectures in core mathematical venues across the globe.
STAGE n+2. A clear Thematic Principle, with immediately striking
resonance, that cuts across virtually all areas of mathematics,
opening up a myriad of related investigations, with unexpected
outcomes relating to existing research.
STAGE n+3. New mathematical tools, based on a clear thematic principle
from stage n+2, that provides necessary uses of large cardinals to
obtain consensus significant theorems across mathematics.

Over the years, I had no confidence that I would ever reach stage n.
Yet we are here, in the middle of stage n, moving steadily along.
Things are getting yet simpler, yet more transparent, and yes, yet
more Perfectly Natural. This is not yet fully played out.

But as a consequence of this, we are seriously approaching Stage n+1,
at least knocking on the door. There is also definitely a small
GLIMPSE at stage n+2. Stage n+3 is still far removed.

NOW THERE is an important distinction here. For mathematicians with
serious Foundational Interests and Sensibilities, they are going to be
already interested in STAGE n statements FOR THEIR
FOUNDATIONAL INTEREST. I.e., that Stage n statements ACTUALLY EXIST,
contrary to CW. THAT is to be distinguished from being
interested purely mathematically.That is stage n+1.

With regard to Stage n+2, we are definitely moving to themes like:

REGULARITY IN MAXIMAL OBJECTS

and similar phrases.

************************************************************************
My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
This is the 791st 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 #1-699 can be found at

700: Large Cardinals and Continuations/14  8/1/16  11:01AM
701: Extending Functions/1  8/10/16  10:02AM
702: Large Cardinals and Continuations/15  8/22/16  9:22PM
703: Large Cardinals and Continuations/16  8/26/16  12:03AM
704: Large Cardinals and Continuations/17  8/31/16  12:55AM
705: Large Cardinals and Continuations/18  8/31/16  11:47PM
706: Second Incompleteness/1  7/5/16  2:03AM
707: Second Incompleteness/2  9/8/16  3:37PM
708: Second Incompleteness/3  9/11/16  10:33PM
709: Large Cardinals and Continuations/19  9/13/16 4:17AM
710: Large Cardinals and Continuations/20  9/14/16  1:27AM
700: Large Cardinals and Continuations/14  8/1/16  11:01AM
701: Extending Functions/1  8/10/16  10:02AM
702: Large Cardinals and Continuations/15  8/22/16  9:22PM
703: Large Cardinals and Continuations/16  8/26/16  12:03AM
704: Large Cardinals and Continuations/17  8/31/16  12:55AM
705: Large Cardinals and Continuations/18  8/31/16  11:47PM
706: Second Incompleteness/1  7/5/16  2:03AM
707: Second Incompleteness/2  9/8/16  3:37PM
708: Second Incompleteness/3  9/11/16  10:33PM
709: Large Cardinals and Continuations/19  9/13/16 4:17AM
710: Large Cardinals and Continuations/20  9/14/16  1:27AM
711: Large Cardinals and Continuations/21  9/18/16 10:42AM
712: PA Incompleteness/1  9/23/16  1:20AM
713: Foundations of Geometry/1  9/24/16  2:09PM
714: Foundations of Geometry/2  9/25/16  10:26PM
715: Foundations of Geometry/3  9/27/16  1:08AM
716: Foundations of Geometry/4  9/27/16  10:25PM
717: Foundations of Geometry/5  9/30/16  12:16AM
718: Foundations of Geometry/6  101/16  12:19PM
719: Large Cardinals and Emulations/22
720: Foundations of Geometry/7  10/2/16  1:59PM
721: Large Cardinals and Emulations//23  10/4/16  2:35AM
722: Large Cardinals and Emulations/24  10/616  1:59AM
723: Philosophical Geometry/8  10/816  1:47AM
724: Philosophical Geometry/9  10/10/16  9:36AM
725: Philosophical Geometry/10  10/14/16  10:16PM
726: Philosophical Geometry/11  Oct 17 16:04:26 EDT 2016
727: Large Cardinals and Emulations/25  10/20/16  1:37PM
728: Philosophical Geometry/12  10/24/16  3:35PM
729: Consistency of Mathematics/1  10/25/16  1:25PM
730: Consistency of Mathematics/2  11/17/16  9:50PM
731: Large Cardinals and Emulations/26  11/21/16  5:40PM
732: Large Cardinals and Emulations/27  11/28/16  1:31AM
733: Large Cardinals and Emulations/28  12/6/16  1AM
734: Large Cardinals and Emulations/29  12/8/16  2:53PM
735: Philosophical Geometry/13  12/19/16  4:24PM
736: Philosophical Geometry/14  12/20/16  12:43PM
737: Philosophical Geometry/15  12/22/16  3:24PM
738: Philosophical Geometry/16  12/27/16  6:54PM
739: Philosophical Geometry/17  1/2/17  11:50PM
740: Philosophy of Incompleteness/2  1/7/16  8:33AM
741: Philosophy of Incompleteness/3  1/7/16  1:18PM
742: Philosophy of Incompleteness/4  1/8/16 3:45AM
743: Philosophy of Incompleteness/5  1/9/16  2:32PM
744: Philosophy of Incompleteness/6  1/10/16  1/10/16  12:15AM
745: Philosophy of Incompleteness/7  1/11/16  12:40AM
746: Philosophy of Incompleteness/8  1/12/17  3:54PM
747: PA Incompleteness/2  2/3/17 12:07PM
748: Large Cardinals and Emulations/30  2/15/17  2:19AM
749: Large Cardinals and Emulations/31  2/15/17  2:19AM
750: Large Cardinals and Emulations/32  2/15/17  2:20AM
751: Large Cardinals and Emulations/33  2/17/17 12:52AM
752: Emulation Theory for Pure Math/1  3/14/17  12:57AM
753: Emulation Theory for Math Logic  3/10/17  2:17AM
754: Large Cardinals and Emulations/34  3/12/17  12:34AM
755: Large Cardinals and Emulations/35  3/12/17  12:33AM
756: Large Cardinals and Emulations/36  3/24/17  8:03AM
757: Large Cardinals and Emulations/37  3/27/17  2:39AM
758: Large Cardinals and Emulations/38  4/10/17  1:11AM
759: Large Cardinals and Emulations/39  4/10/17  1:11AM
760: Large Cardinals and Emulations/40  4/13/17  11:53PM
761: Large Cardinals and Emulations/41  4/15/17  4:54PM
762: Baby Emulation Theory/Expositional  4/17/17  1:23AM
763: Large Cardinals and Emulations/42  5/817  2:18AM
764: Large Cardinals and Emulations/43  5/11/17  12:26AM
765: Large Cardinals and Emulations/44  5/14/17  6:03PM
766: Large Cardinals and Emulations/45  7/2/17  1:22PM
767: Impossible Counting 1  9/2/17  8:28AM
768: Theory Completions  9/4/17  9:13PM
769: Complexity of Integers 1  9/7/17  12:30AM
770: Algorithmic Unsolvability 1  10/13/17  1:55PM
771: Algorithmic Unsolvability 2  10/18/17  10/15/17  10:14PM
772: Algorithmic Unsolvability 3  Oct 19 02:41:32 EDT 2017
773: Goedel's Second: Proofs/1  Dec 18 20:31:25 EST 2017
774: Goedel's Second: Proofs/2  Dec 18 20:36:04 EST 2017
775: Goedel's Second: Proofs/3  Dec 19 00:48:45 EST 2017
776: Logically Natural Examples 1  12/21 01:00:40 EST 2017
777: Goedel's Second: Proofs/4  12/28/17  8:02PM
778: Goedel's Second: Proofs/5  12/30/17  2:40AM
779: End of Year Claims  12/31/17  8:03PM
780: One Dimensional Incompleteness/1  1/4/18  1:14AM
781: One Dimensional Incompleteness/2  1/6/18  11:25PM
782: Revolutionary Possibilities/1  1/12/18  11:26AM
783: Revolutionary Possibilities/2  1/20/18  9:43PM
784: Revolutionary Possibilities/3  1/21/18  2:59PM
785: Revolutionary Possibilities/4  1/22/18  12:38AM
786: Revolutionary Possibilities/5  1/24/18  12:15AM
787: Revolutionary Possibilities/6  1/25/18  4:09AM
788: Revolutionary Possibilities/7  2/1/18  2:18AM
789: Revolutionary Possibilities/8  2/1/18  9:02AM
790: Revolutionary Possibilities/9  2/2/18  3:07AM

Harvey Friedman
```