886: Concerning Natural/1
Harvey Friedman
hmflogic at gmail.com
Sun May 16 02:00:36 EDT 2021
Issues and controversies and conundrums abound in f.o.m. concerning
the attribute "natural". Of course this also occurs outside f.o.m. as
well.
There is a real need for some better and more comprehensive
understanding of what is meant by natural - at least how we use the
general concept of NATURAL in f.o.m. research.
WHERE NATURALNESS IS CRITICAL IN F.O.M.
DEFENSES AGAINST SHOCKS
STRENGTHENING SHOCKS
Naturalness has become particularly critical in certain focused f.o.m.
contexts. Here are some.
1. Nonrecursive sets of integers.
2. Degrees between 0 and 0'.
3. Ordinal notation systems.
4. Formal systems.
5. Sentences neither provable nor refutable in formal systems.
IIn many of these cases, the issue of naturalness arises because there
is an initial STARTLING DEVELOPMENT establishing the certain phenomena
in mathematics exist. Often these phenomena were at least silently
conjectured to NOT EXIST. So when the phenomena gets uncovered, there
is a certain amount of SHOCK VALUE in the mathematical community. And
when there is a SHOCK, there are two kinds of natural (!!) reactions:
A. Great excitement to look for more related SHOCKS, especially ones
that are even more shocking.
B. Defense against SHOCKS. I.e., to justify the claim that people
shouldn't have been so shocked in the first place.
B involves pointing out features that the so called "shocking"
examples have that are NOT present in WHAT NATURE HAS ALREADY CREATED.
Now there is no question that we can generally expect a lot of
examples of various things in NATURE over the next MILLION years that
we have not seen seen in NATURE up to now.
But it might be claimed that the SHOCKING examples have this or that
in them but up to now, in NATURE, we don't see this or that --- And
maybe it is plausible that in a MILLION years one would not see this
or that in nature.
So this suggests the following SHOCK REINFORCEMENT, instilling STRONGER SHOCKS.
I. Develop examples and surrounding developments, that are so
compelling in their "intrinsic interest" and "intrinsic beauty" and
"explanatory vision" and "mathematical depth" and "thematic character"
as compared to various celebrated mathematical developments already
occurring in nature, that it is clear that either they were INEVITABLE
to occur in Nature over the next million years, or if they didn't,
that would be a great DISAPPOINTMENT in what is going to happen in the
next million years.
II. Argue specifically for their INEVITABILITY, based on their
THEMATIC CHARACTER. I.e., that they straightforwardly combine existing
ideas and constructions that are well known and already in Nature,
that had not been combined before.
Item II needs to be taken seriously because there are quite a number
of absolutely fundamentally important well accepted well known ideas
in mathematics that HAVE NOT BEEN COMBINED IN ANY MEANINGFUL WAY,
often because the experts in them are from different branches of
mathematics and the cross culture is just not that strong or
systematic.
BRIEF OVERVIEW OF STATE OF THE ART IN THOSE 5 ABOVE
1. Nonrecursive sets of integers. Progress on this is actually quite
limited. There are some weak forms of this that definitely had SOME
SHOCK value when they first came out, but DEFENSES have been built up.
Most famously, we know from MRDP that there are polynomials with
integer coefficients and integer variables whose range is a non
recursive set of integers. Shocking, at least to some extent, in the
1970's. But the DEFENSE is that "this construction is not of a
specific set of integers but one only relative to a choice of
polynomial" and also "polynomials that occur in nature are of small
degree with a small number of variables". For making both small degree
and small number of variables, the state of the art is roughly mid
20's for both simultaneously. But it is open even for degree 3 and two
variables. There has been no major progress in this for well over 40
years.
We can of course go way beyond polynomials here and consider GENERAL
MATHEMATICAL EXPRESSIONS with integer values. But here, because of the
use of various operations, for SHOCK one needs the expression to be
rather simple and small. Or it could be not like that at all, but
instead VERY CONCEPTUAL. An attempt at this is made in
(with T. Erdelyi), The Number of Certain Integral Polynomials and
Nonrecursive Sets of Integers, Part I, Transactions of the AMS, 357,
(2005), 999-1011.
The Number of Certain Integral Polynomials and Nonrecursive Sets of
Integers, Part II, Transactions of the AMS, 357 (2005), 1013-1023.
Not at all clear to me where to take this much further - or better.
2. Degrees between 0 and 0'. I don't know of any progress whatsoever
on this. I am also not aware of any progress on showing that there is
no natural degree between 0 and 0'.
3. Ordinal Notation Systems. .It seems totally convincing that the
usual ordinal notation systems have some special naturalness. The
issue arises particularly because we know from recursion theory that
well orderings of a given even modest ordinal may be not even be
recursively embeddable in each other, let alone recursively
isomorphic. The ideal result would be some condition to be placed on
the well orderings of omega of order type that ordinal are all
recursively isomorphic, or even better low computationally isomorphic.
I think that this is a viable aim for the ordinals of Pi11-CA0. I will
be checking some ideas with experts in ordinal notations soon.
4. Formal Systems. Of course, the whole idea of my RM is that various
Formal Systems are provably equivalent to various mathematical
fundamentally interesting and important mathematical statements, and
this can be highly satisfactory. But this isn't readily carried out
except for many subsystems of Z_2. It has been carried out for
Z_2,Z_3,...,Z_alpha for alpha a naturally given recursive ordinal, by
using levels of Borel Determinacy, Borel Selection, or Borel
Diagonalization, but only really for their logical strength or their
provable Pi12 or Pi13 sentences. Actually one should examine the
meaning of this more carefully especially in the sense that it
characterizes the provable Pi01, Pi02, Pi11 theorems of these
theories.
I have some extremely strong, in my opinion, work characterizing ZF
and beyond in terms of going from the finite to the infinite. I have
posted on this on the FOM but it is a very much overdue major project
I haven't rolled out properly although I do remember talking on it at
UPenn many years ago at an ASL meeting.
5. Sentences neither provable nor refutable in formal systems. This is
of course my main agenda these days and for 54 years. And I have a lot
to say about how this has gone over the last 54 years including work
of others, and the milestones and prospects, and how we can tell
whether things are natural and how natural and what affect it does and
what affect it should and what affect it will have on the mathematics
community. Everybody knows that it has its ultimate origins in
Goedel's Incompleteness Theorems. More later, of course. One limited
corner of all this is discussed in #885: Effective Forms.
##########################################
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 886th 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
884: Low Strength Zoo/1
885: Effective Forms
Harvey Friedman
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