884: Low Strength Zoo/1

Sun May 16 01:34:57 EDT 2021

The issue of the linear ordering of logical strengths that has been
discussed in https://cs.nyu.edu/pipermail/fom/2021-May/022649.html and
the immediately earlier postings on this by others here on FOM. The
takeaway for me is that we see a profound linearity of
INTERPRETABILITY among obviously natural theories if at least one
interprets EFA - with counterexamples that exhibit admittedly *some
sort of* naturality. But I maintain that there is some idea of "FOUND
IN NATURE" that makes 100% robust sense in the development of RM and
the RM ZOO, for which one does (seem to have) linearity. That if one
pushes the standards up for naturalness in this way, there is are no
counterexamples in sight. The RM ZOO has a clear methodology of
development, and it does appear that it is not reaching anything like
a breakdown in the linear ordering under interpretability let alone
provable implications between consistency statements say over PRA.

But here I want to talk about something ENTIRELY DIFFERENT. What if we
are talking about theories that are WEAK, that lie below the threshold
of Goedel Incompleteness? And what should we mean by this threshold?

Actually, there is an enormous proliferation of totally natural
theories - theories arising in nature - that live below or well below
Goedel Incompleteness. Some of these are studied by logicians for
reasons having nothing to do with Incompleteness and traditional
issues in the foundations of mathematics. Perhaps most notably, these
are studied by model theorists often but not always in connection with
ongoing mainstream mathematical practice, with an eye towards
applications or at least simplifications and consolidations of
existing mathematical developments.

We see a HUGE AMOUNT OF INCOMPARABILITY here among theories of "no
logical strength" or "low logical strength". I am proposing a
systematic study of these theories under INTERPRETABILITY, a decently
robust notion normally credited to Alfred Tarski.

HOWEVER, there are some robustness issues connected with the notion of
Interpretability and they have been investigated some. Probably one
should use the most liberal notion of interpretation that immediately
gives relative consistency. Take a look at the paper

When Bi-Interpretability Implies Synonymy
Visser, Albert; Friedman, Harvey M.
(2014) Logic Group preprint series, volume 320

with the Appendices and the references.

I.e., I am proposing the systematic development of the LOW STRENGTH
ZOO. I am totally open to other proposed names for this.

When I have talked about "having logical strength" in the past, I
normally have identified this with INTERPRETING EFA.

NOTIONS OF HAVING STRENGTH

1. Interpreting EFA.
2. Interpreting Robsinson's Q.
3. Interpreting AST (adjunctive set theory).
4. Interpreting Presburger Arithmetic.
5. Interpreting Paring Function.
6. All models non recursive.
7. Set of theorems not recursive.

It is not clear to me just what should be used here in a systematic
informative development of LOW STRENGTH ZOO - one of these or
something else. But nevertheless I see these particular ZOO Areas:

A. Fragments of real closed field axioms.
B. Fragments of algebraically closed field axioms.
C. Fragments of Presburger Arithmetic.
D. Fragments of Euclidean Geometry (of course like A but in a
different language suggesting different Zoo animals).
E. Various extensions of the linear order axioms.
F. Various extensions of the Boolean algebra axioms.
G. Various extensions of the group, field, ring axioms.
E. Fragments of the linear order axioms. (not necessarily literal fragments)
F. Fragments of the group axioms (not necessarily literal fragments)
G. Fragments of the field axioms (not necessary literal fragments)

where we expect massive amounts of NONLINEARITY under
Interpretability, among extremely natural cases.

##########################################

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 884th 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
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
834: Tangible Incompleteness Restarted/8  10/10/19  5:02PM
835: Tangible Incompleteness Restarted/9  10/13/19  4:50AM
836: Tangible Incompleteness Restarted/10  10/14/19  12:34PM
837: Tangible Incompleteness Restarted/11 10/18/20  02:58AM
838: New Tangible Incompleteness/1 1/11/20 1:04PM
839: New Tangible Incompleteness/2 1/13/20 1:10 PM
840: New Tangible Incompleteness/3 1/14/20 4:50PM
841: New Tangible Incompleteness/4 1/15/20 1:58PM
842: Gromov's "most powerful language" and set theory  2/8/20  2:53AM
843: Brand New Tangible Incompleteness/1 3/22/20 10:50PM
844: Brand New Tangible Incompleteness/2 3/24/20  12:37AM
845: Brand New Tangible Incompleteness/3 3/28/20 7:25AM
846: Brand New Tangible Incompleteness/4 4/1/20 12:32 AM
847: Brand New Tangible Incompleteness/5 4/9/20 1 34AM
848. Set Equation Theory/1 4/15 11:45PM
849. Set Equation Theory/2 4/16/20 4:50PM
850: Set Equation Theory/3 4/26/20 12:06AM
851: Product Inequality Theory/1 4/29/20 12:08AM
852: Order Theoretic Maximality/1 4/30/20 7:17PM
853: Embedded Maximality (revisited)/1 5/3/20 10:19PM
854: Lower R Invariant Maximal Sets/1:  5/14/20 11:32PM
855: Lower Equivalent and Stable Maximal Sets/1  5/17/20 4:25PM
856: Finite Increasing reducers/1 6/18/20 4 17PM :
857: Finite Increasing reducers/2 6/16/20 6:30PM
858: Mathematical Representations of Ordinals/1 6/18/20 3:30AM
859. Incompleteness by Effectivization/1  6/19/20 1132PM :
860: Unary Regressive Growth/1  8/120  9:50PM
861: Simplified Axioms for Class Theory  9/16/20  9:17PM
862: Symmetric Semigroups  2/2/21  9:11 PM
863: Structural Mapping Theory/1  2/4/21  11:36PM
864: Structural Mapping Theory/2  2/7/21  1:07AM
865: Structural Mapping Theory/3  2/10/21  11:57PM
866: Structural Mapping Theory/4  2/13/21  12:47AM
867: Structural Mapping Theory/5  2/14/21  11:27PM
868: Structural Mapping Theory/6  2/15/21  9:45PM
869: Structural Proof Theory/1  2/24/21  12:10AM
870: Structural Proof Theory/2  2/28/21  1:18AM
871: Structural Proof Theory/3  2/28/21  9:27PM
872: Structural Proof Theory/4  2/28/21  10:38PM
873: Structural Proof Theory/5  3/1/21  12:58PM
874: Structural Proof Theory/6  3/1/21  6:52PM
875: Structural Proof Theory/7  3/2/21  4:07AM
876: Structural Proof Theory/8  3/2/21  7:27AM
877: Structural Proof Theory/9  3/3/21  7:46PM
878: Structural Proof Theory/10  3/3/21  8:53PM
879: Structural Proof Theory/11  3/4/21  4:22AM
880: Tangible Updates/1  4/15/21 1:46AM
881: Some Logical Thresholds  4/29/21  11:49PM
882: Logical Strength Comparability  5/8/21 5:49PM
883: Tangible Incompleteness Lecture Plans

Harvey Friedman