882: Logical Strength Comparability

Sat May 8 17:49:12 EDT 2021

FOM readers may not realize this, but I wrote

881: Some Logical Thresholds
https://cs.nyu.edu/pipermail/fom/2021-April/022635.html

inspired by discussion of Incomparable Logical Strengths. The
suggestion there is that in terms of crude complexity, the place where
you get a lot of incomparable statements is NOT the place where you
get incomparable logical strengths.

Let us consider the Reverse Mathematics Zoo:

https://rmzoo.math.uconn.edu/#

The following has 11 fascinating deep links:

https://rmzoo.math.uconn.edu/diagrams/

Also

https://projecteuclid.org/journals/notre-dame-journal-of-formal-logic/volume-59/issue-4/Refining-the-Taming-of-the-Reverse-Mathematics-Zoo/10.1215/00294527-2018-0015.short

http://www.ludovicpatey.com/media/research/zoo.pdf

https://www.computability.org/zoo-viewer/

https://www.birs.ca/cmo-workshops/2019/19w5111/Report19w5111.pdf

https://www.birs.ca/events/2019/5-day-workshops/19w5111/videos

Note that in the entire Reverse Mathematics Zoos (the last reference
above is a less developed newer Zoo region), ALL OF THE LOGICAL
STRENGTHS ARE COMPARABLE, as far as we know.

RHETORICAL QUESTION: Where is the comparable Zoo of Logical Strengths
in Mathematics?

So the COMPARABILITY of LOGICAL STRENGTHS is definitely a pervasive
phenomena where the EXCEPTIONS ARE AWKWARD.

We know how Reverse Mathematics is conducted and developed. In the
course of that development, we have now very natural developments,
rather systematic, of WILD ZOOS of incomparables.

CONJECTURE. There is no systematic development of Logical Strength Zoo.

CONJECTURE. There is a theoretical reason for no systematic Logical
Strength Zoo as there is a Systematic RM Zoo, and this theoretical
reason is deep and worth uncovering.

These so called "counterexamples" to comparability just show how deep
an idea is needed to analyze this apparent Comparability. In no way a
reason to lessen one's great excitement over the Apparent
Comparability.

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

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 882nd 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

Harvey Friedman