880: Tangible Updates/1
Harvey Friedman
hmflogic at gmail.com
Thu Apr 15 01:46:56 EDT 2021
I started a Zoom lecture series which is still at the experimental
stage before we open it up to a wide audience. I give the planned
weekly lecture on Zoom, it is recorded and Archived, and then very
shortly thereafter I revise and expand the underlying notes and put
the revised and expanded version on my webpage BUT UNDER DOWNLOADABLE
LECTURE NOTES. This is different than my usual Downloadable Manuscript
page.
This Lecture Note page is at
https://cpb-us-w2.wpmucdn.com/u.osu.edu/dist/1/1952/files/2021/04/TangibleIncGent041421.pdf
and the first installment is
70. Tangible Incompleteness Series, opening lecture, Zoom, University
of Gent, Gent, Belgium. April 12, 2021. 15 pages. Revised and expanded
April 14, 2021.
These Lecture Notes are considerably less polished than what appears
on my Downloadable Manuscripts Page, which are often very polished by
my standards.
I have been absent for a while finishing up a lot of the Tangible
Incompleteness book MINUS reversals, which is still considerable work.
Forward directions and all machinations thereof is still pretty
intricate. There have been a great deal of major improvements, a lot
of added material and so forth. Especially with two essentially
completed Classification Projects showing that the invariance
conditions I use in Invariant Maximality are best possible. This
constitutes a kind of proof that this realm of Tangible Incompleteness
is completely natural.
Above all, there has been major advances in the explicitly Pi01 forms.
The highlight of this is a simple nondeterministic algorithm which can
be carried out for infinitely many steps if and only if the HUGE
cardinal hierarchy is consistent. This is of course the same as the
algorithm being carried out for any finite number of steps. The
setting is the rationals, and so this is explicitly Pi02. However, it
is trivial from decision procedures or just directly seeing an upper
bound on the numerators and denominators of the rationals required,
that it is explicitly Pi01.
Furthermore, these nondeterministic algorithms are very simple to
present compared to what is standard in computer science theory.
I am going to roll out all of this and more in the weekly Zoom
lectures, and also talk about some of it here.
Also with the Zoom lectures, the idea is that I am going to LET MY
HAIR DOWN and talk expansively about foundational issues.
The Tangible Incompleteness Interim Report update is very well
polished and part of it will be done shortly on INVARIANT MAXIMALITY -
DERIVATIONS. This will be placed on my regular Downloadable
Manuscripts page, not the Lecture Note page.
As I have said earlier on FOM, there has been an unexpected "proof" of
the total naturalness and fundamental character of the Incompleteness.
This is because of the apparent total suitability of a form of the
theorems for GIFTED HIGH SCHOOL STUDENTS. My discussions with the
planned interaction with them is ongoing and EXTREMELY ENCOURAGING.
The real interaction will be live in June, 2022. Probably in this
situation there can only be limited success with this audience through
online interactions only. But we shall see.
MORE SPECIFICALLY. There is one form of the implicitly Pi01 (not
explicitly Pi01) that takes the form
*For all k,n >= 1, every finite E containedin Q[-n,n]^k has a maximal
emulator with a simple invariance condition"
I know this is equivalent to Con(SRP). And probably is beyond ZFC for
k = 3 and even n = 3 (maybe). In dimension 2 for arbitrary n, my
sledgehammer proves this just beyond Z_2. But my sledgehammer is
highly nonconsecutive even in that the invariant maximal emulators are
generally not recursive.
HOWEVER, here is the baby case so perfect for the students at this level:
*Every set E containedin Q[-1,1]^2 of cardinality at most 3 has a
maximal emulator with a simple invariance condition"
where now in 2 dimensions, the simple invariance condition is really
TRIVIAL to state (not just simple) for students at this level.
The plan is to walk them through a complete proof of this with "3".
And they get an EFFECTIVELY COMPUTED maximal emulator with the trivial
invariance condition. That is something I don't get when I use my
professional sledgehammer. So the students can immediately get
involved in pushing the state of the art right away.
I will close this HELLO posting with the current table of contents for
the Invariant Maximality part of the book.
2.1. INVARIANT MAXIMALITY - DERIVATIONS
2.1.1. Introduction
2.1.2. Preliminaries, Equivalence Relations, Invariance, Seven
2.1.3. Headline Results: Maximal Squares
2.1.4. r-Cubes, r-Sides, r-Cliques, r-Emulators
2.1.5. Inspirational Classification Projects
2.1.5.1. EQR on Jk Involving <
2.1.5.2. EQR on Qk Involving <,Z
2.1.5.3. EQR on Qk Involving <,Z,+
2.1.5.4. 35 Project, ZOI Project
2.1.5.5. Classification Projects
2.1.5.6. OT Project in [Fr17]
2.1.6. Outliers
2.1.6.1. OE
2.1.6.2. ZOE
2.1.6.3. k = 1
2.1.6.4. r = 1
2.1.6.5. ME iff MG*
2.1.7. OE/Z*
2.1.7.1. The Nine Types of Intervals
2.1.7.2. (no,no),(no,yes),(no,inf),(yes,no),(-inf,no)
2.1.7.3. (yes,yes)
2.1.7.4. Witness Systems and (yes,inf),(inf,yes),(-inf,inf)
2.1.7.5. OE/Z* Comprehensive
2.1.8. OE/upZ<=0
2.1.8.1. Greedy Transfinite Construction
2.1.8.2. OE/upZ<=0
2.1.9. OE/upZ<0
2.1.10. OE/upZ<=Z
2.1.11. OE/upZ<Z
2.1.12. 35 Project - Comprehensive
2.1.13. 35 Project - Provability/Unprovability
2.1.14. The ZOI Project - Two Dimensions
2.1.15. The ZOI Project - OE/upZ<Z is Maximum
2.1.16. r-Duplicators and Finite Seeds
2.1.17. Detailed Metamathematical Features
2.1.18. Explicitly Finite Forms
2.1.18.1. Logic Based
2.1.18.2. Finite Approximation Towers
2.1.18.3. Nondeterministic Algorithms
2.1.19. Emulators in Q[-1,1]^2 and Mathematically Gifted Youth
Appendix A: The Stationary Ramsey Property
Appendix B: The Strongly Mahlo Hierarchy
Appendix C: Formal Systems Used
References
##########################################
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 878th 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
Harvey Friedman
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