887: Updated Adventures
Harvey Friedman
hmflogic at gmail.com
Thu Sep 9 09:47:16 EDT 2021
I have been tied up with a number of obligations and initiatives that
have arisen with my Gent Lectures and the two recent meetings on
Goedel's Incompleteness Theorems, the one in Tubingen and the later
one in Wuhan:
The completed version of my talk is now at
Aspects of Goedel Incompleteness, posting #78, Downloadable Lecture
Notes, September 8, 2021,
https://u.osu.edu/friedman.8/foundational-adventures/downloadable-lecture-notes-2/
The above is the Downloadable Lecture Notes page, not the Downloadable
Manuscripts page.
There is quite a bit of material on Goedel Incompleteness in the
reasonably polished #78 with proofs of main new results obtained for,
during, and after that splendid productive meeting.
The time has come for the detailed presentation of the reversals at
the Gent Lectures. This is very brittle in the sense that if it is not
done completely correctly then one has to make major changes in the
setup and previous lectures have essentially to be discarded and
redone. This is totally unacceptable for a Zoom Lecture Series.
Consequently, I decided to wait till I have a complete manuscript
completely checked by me, and then lecture from that.
CONCERNING THE REVERSAL BEING PREPARED for GENT LECTURES
Look at these statements which lead the general mathematician down the
garden path to Large Cardinals.
1. Every order invariant subset of Q[0,n]2k has a maximal square.
This is a nice undergraduate exercise. Proved by an obvious greedy
algorithm in RCA0 with a low level computational square. The square
here refers to some S x S contained in the given order invariant
subset of Q[0,n]2k.
MOTIVATING TOPIC: How nice can be require the maximal square to be?
I.e., "the theory of maximal squares in order invariant sets". Or
"order invariant maximal square theory".
2. Every order invariant subset of Q[0,n]2k has an order invariant
maximal square.
Order invariance is extremely mathematically natural and extremely
strong. 2 if true would be the be all and end all of all of this
stuff. I.e., of the theory of maximal squares in order invariant sets.
UNFORTUNATELY, 2 is extremely false, with a totally understandable
strong refutation.
LET US attempt to RECOVER from this disappointment.
3. Every order invariant subset of Q[0,n]2k has a maximal square which
is order invariant over Z[0,n].
THus instead of requiring order invariance (of the maximal square), we
instead only require that this order invariance work over the
INTEGERS. A very very natural move.
UNFORTUNATELY, 3 is refutable.HOWEVER, the refutation is nontrivial
and quite interesting. The ANNOYING endpoints are the problem here. So
let's get rid of one or both of the endpoints.
4. Every order invariant subset of Q[0,n]2k has a maximal square which
is order invariant over Z[1,n-1].
Yes this is fine and easy for those who like the classical finite
Ramsey theorem. The relevant homogeneous set is going to be
{1,...,n-1}.
5. Every order invariant subset of Q[0,n]2k has a maximal square which
is order invariant over Z[1,n].
Every order invariant subset of Q[0,n]2k has a maximal square which is
order invariant over Z[0,n-1].
These are obviously equivalent by reversing the order. THESE are real
mathematical theorems, that can be proved with some clever ideas and
the proofs do not involve uncountable sets. The proofs use a powerful
application of the classical infinite Ramsey theorem. Thus this is
like 4 only much more difficult.
6. Every order invariant subset of Q[0,n]2k has a maximal square whose
<1 sections are order invariant over Z[1,n].
HERE we have used a rather common mathematical construction. The
sections of a higher dimensional set are obtained by fixing zero or
more of the coordinates by elements and taking the lower dimensional
reduction. Here we take only the <1 sections, which just means that
the fixed coordinates must lie in Q[0,1). Obviously by suing the empty
section, 6 certainly implies 5.
WOW: 6 is independent of ZFC. 6 is provably equivalent to Con(SRP) over WKL0.
Earlier Gent lectures (almost written up in final form) proves 6 (and
stronger) in WKL0 + Con(SRP). In fact, they proved the stronger form
of 6:
7. Every order invariant subset of Q[0,n]2k has a maximal square
whose <i sections, i < n, are order invariant over Z[i,n].
So 7 is also provably equivaelnt to Con(SRP) over WKL0.
##########################################
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 887th 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 5/16/21 1:29:44
884: Low Strength Zoo/1 5/16/21 1:34:
885: Effective Forms 5/16/21 1:47AM
886: Concerning Natural/1 5/16/21 2:00AM
Harvey Friedman
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