883: Tangible Incompleteness Lecture Plans

Sun May 16 01:29:44 EDT 2021

I have been moving towards the idea of disseminating my research and
research ideas through recordings and associated Lecture Notes but I
still would like to maintain a dialog with scholars on the FOM.

I have committed myself to weekly recorded Lectures on Zoom organized
at Gent for the indefinite future. And for the indefinite future, it
will be rolling out all of my work on Tangible Incompleteness there.
Things are going well, with 5 recordings already and five Lecture
Notes manuscripts on my Downloadable Manuscripts webpage.
https://u.osu.edu/friedman.8/foundational-adventures/downloadable-lecture-notes-2/

Lecture 6 will be May 19 at 10AM EST, contact A. Weiermann. In Lecture
5, I have a complete proof of

Q[-1,1]^k EMULATION THEOREM. Every subset of Q[-1,1]^k has an ush
stable maximal r-emulator.

using transfinite recursion on omega_1 x omega. Gifted high school
lives in fragments of k = 2 and r = 2. They are walked through <= 3
element subsets of Q[-1,1]^2, where no transfinite machines are
needed.

So I am in the middle of proofs of the new Stable Maximality. I
changed the name from Invariant Maximality to Stable Maximality
because stability is what drives everything, and stability is the same
as invariance except with "if and only if", sometimes called
"completely invariant". If we are talking about an equivalence
relation, then invariance and stability are obviously the same. Hence
I use "order invariance" a lot and it is the same as "order
stability".

So during this period of proofs of Stable Maximality statements, I
sharpen up the statements some by using r-sides. The proof is the
same:

Q[-1,1]^kr SIDES THEOREM. Every order invariant subset of Q[-1,1]^kr
has an ush stable maximal r-side.

An r-side in E containedin Q[-1,1]^kr is an S containedin Q[-1,1]^k
with S^r containedin E.

Notice that the use of order invariance has now been moved up front in
the statement.

We next move to this:

Q[-2,2]^4 SIDES*. Every order invariant subset of Q[-2,2]^4 has an ush
stable maximal 2-side.

* But there is a problem with using ush. In ush on Q[-1,1] we only
move 0 to 1 because moving anything larger would throw us out of the
space. With Q[-2,2] we move all of Q[0,1] onto all of Q[1,2]. This is
quite different. And too strong.

THEOREM. Q[-2,2]^4 Sides is false. It is refutable in RCA_0.

What do we do about this? Well we really only want to move nonnegative
integers. So we define the N tail of x in Q^k as the x_i such that
every x_j >= x_i lies in N. Then we use the N tail shift, or Ntsh,
instead of the upper shift, or ush.

The strongest of such stability is given by the closely related "N
tail related" equivalence relation on Q^k. This means the tuples are
order equivaelnt and their N tails occupy exactly the same positions.
We will eventually prove that "N tail related stability" is the
strongest stability we can use for our statements, in an appropriate
sense.

Q[-n,n]^2r SIDES. Every order invariant subset of Q[-n,n]^2r has an
Ntsh stable maximal r-side.
Q[-n,n]^2r SIDES. Every order invariant subset of Q[-n,n]^2r has an
Ntrel stable maximal r-side.

We also give a proof of this using transfinite recursion on omega_1 x omega.

In Lecture 7 on May 26, we start by discussing the fact that all of
the statements we have been considering, and will consider for some
time, are explicitly Pi01 over WKL_0, with an explanation of what that
means, and also that they are "demonstrably falsifiable", something
that is almost imperative in physical science. We show that they are
explicitly Pi01 by a basic argument using Goedel's Completeness
Theorem.

We then take the  plunge into

Q[-n,n]^3r SIDES. Every order invariant subset of Q[-n,n]^3r has an
Ntrel stable maximal r-side.

We try to use the same proof that we used on Q[-n,n]^2r but we run
into trouble needing combinatorial properties of omega_1 that just
aren't true. We replace omega_1 by a subtle cardinal to push this
through.

So we need to discuss the SRP hierarchy in some detail before going
further with the proof

Q[-n,n]^kr SIDES. Every order invariant subset of Q[-n,n]^kr has an
Ntrel stable maximal r-side.

using the entire SRP hierarchy, in Lecture 8 on June 2. Then we
consider weakenings of Ntrel stable so that we can get away with the
entire strongly Mahlo cardinal hierarchy, and also with the first
omega infinite cardinals.

The plan is to begin rolling out the reversals in Lecture 9 on June 9.
The first reversal will be to show that Q[-n,n]^kr SIDES above implies
Con(SRP). The proof given in Lecture 8 is shown to be adapted to a
proof in WKL_0 + Con(SRP).

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

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 883rd 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

Harvey Friedman