[FOM] 691:Large Cardinals and Continuations/10
Harvey Friedman
hmflogic at gmail.com
Mon Jun 13 21:09:23 EDT 2016
We have managed to replace the k-dimensional rectangles
Q^k|<0 x {(0,...,k-1)}
Q^k|<0 x {(1,...,k)}.
in Maximal Continuations (MC) by the rectangles
Q|<0 x {(0,...,k-1)}
Q|<0 x {(1,...,k)}.
which are just LINES.
NOTE: In http://www.cs.nyu.edu/pipermail/fom/2016-June/019905.html I
labeled MC as SMC, which was an oversight..
This calls for a redesign of MC.
DEFINITION. Let x,y in Q^k. OL(x,y) is the open line segment of
rational points from x to y.
(REDESIGNED) MAXIMAL CONTINUATION. MC. For finite subsets of
Q[-k,0)^k, some maximal k-continuation within Q[-k,k]^k translates
from OL((0,1,...,k-1), (1,1,...,k-1)) onto OL((0,2,...,k),(1,2,...,k)).
Here (x,y) is the open line segment of rational points from x in Q^k
to y in Q^k.
The proof from SRP+ given in
http://www.cs.nyu.edu/pipermail/fom/2016-June/019905.html trivially
adapts to this new MC, using D instead of D'.
The seed to plant story of course remains unchanged.
The main point is that now the symmetry involved is extremely simple.
I see only one obvious blockbuster improvement left for MC. In the
extreme, this would be, e.g.,
*For finite subsets of Q[-3,0)^3, some maximal 3-continuation within
Q[-3,3]^3 translates from OL((0,1,2),(1,1,2)) onto OL(0,2,3),(1,2,3))*
However, I won't attempt to control numerical parameters like this
until I have a full writeup of the reversal of the redesigned MC
above.
The finite form is as expected:
(REDESIGNED) FINITE RICH CONTINUATION. FRC. For finite subsets of
Q[-k,0)^k, some two successive finite rich k-continuations within
Q[-k,k]^k translate from OL((0,1,...,k-1), (1,1,...,k-1)) onto
OL((0,2,...,k),(1,2,...,k)).
THEOREM. MC is provably equivalent to Con(SRP) over WKL_0. FRC is
provably equivalent to Con(SRP) over EFA.
This move to translations from one line onto another has considerable
repercussions for smooth and strongly smooth continuations. As of now,
for these kinds of continuations, we work in Q^k. In light of these
new developments it is probably best to also work in Q[-k,k]^k. For
smooth continuations, and translate from one given square INTO
another. For strongly smooth continuations, we have a way to translate
from both one given line INTO another and one given line ONTO
another. The former is equivalent to Con(SRP) and the latter is
equivalent to Con(HUGE). FURTHERMORE, in the smooth and strongly
smooth continuation environment, coding techniques are more powerful,
and hence the prospects for dramatic dimension reduction are very good
- better than they are for just pure continuations discussed above.
AND, of course, all of this should be easily finitized as above using
"richness", referring to the use of heights (of rationals, rational
tuples, and sets of rational tuples)
But I am really now pushing past the outer limits of what I can
confidently claim with high probability right now.
So I think I have come to the point where the reversal manuscript
should to be completed. Of course, I said this before, and keep
finding truly major advances. Barring a truly major overhaul, I will
present a revised comprehensive statement and then plunge into
writing.
Harvey Friedman
***********************************************
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 691st 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 #1-599 can be found at the FOM posting
http://www.cs.nyu.edu/pipermail/fom/2015-August/018887.html
600: Removing Deep Pathology 1 8/15/15 10:37PM
601: Finite Emulation Theory 1/perfect? 8/22/15 1:17AM
602: Removing Deep Pathology 2 8/23/15 6:35PM
603: Removing Deep Pathology 3 8/25/15 10:24AM
604: Finite Emulation Theory 2 8/26/15 2:54PM
605: Integer and Real Functions 8/27/15 1:50PM
606: Simple Theory of Types 8/29/15 6:30PM
607: Hindman's Theorem 8/30/15 3:58PM
608: Integer and Real Functions 2 9/1/15 6:40AM
609. Finite Continuation Theory 17 9/315 1:17PM
610: Function Continuation Theory 1 9/4/15 3:40PM
611: Function Emulation/Continuation Theory 2 9/8/15 12:58AM
612: Binary Operation Emulation and Continuation 1 9/7/15 4:35PM
613: Optimal Function Theory 1 9/13/15 11:30AM
614: Adventures in Formalization 1 9/14/15 1:43PM
615: Adventures in Formalization 2 9/14/15 1:44PM
616: Adventures in Formalization 3 9/14/15 1:45PM
617: Removing Connectives 1 9/115/15 7:47AM
618: Adventures in Formalization 4 9/15/15 3:07PM
619: Nonstandardism 1 9/17/15 9:57AM
620: Nonstandardism 2 9/18/15 2:12AM
621: Adventures in Formalization 5 9/18/15 12:54PM
622: Adventures in Formalization 6 9/29/15 3:33AM
623: Optimal Function Theory 2 9/22/15 12:02AM
624: Optimal Function Theory 3 9/22/15 11:18AM
625: Optimal Function Theory 4 9/23/15 10:16PM
626: Optimal Function Theory 5 9/2515 10:26PM
627: Optimal Function Theory 6 9/29/15 2:21AM
628: Optimal Function Theory 7 10/2/15 6:23PM
629: Boolean Algebra/Simplicity 10/3/15 9:41AM
630: Optimal Function Theory 8 10/3/15 6PM
631: Order Theoretic Optimization 1 10/1215 12:16AM
632: Rigorous Formalization of Mathematics 1 10/13/15 8:12PM
633: Constrained Function Theory 1 10/18/15 1AM
634: Fixed Point Minimization 1 10/20/15 11:47PM
635: Fixed Point Minimization 2 10/21/15 11:52PM
636: Fixed Point Minimization 3 10/22/15 5:49PM
637: Progress in Pi01 Incompleteness 1 10/25/15 8:45PM
638: Rigorous Formalization of Mathematics 2 10/25/15 10:47PM
639: Progress in Pi01 Incompleteness 2 10/27/15 10:38PM
640: Progress in Pi01 Incompleteness 3 10/30/15 2:30PM
641: Progress in Pi01 Incompleteness 4 10/31/15 8:12PM
642: Rigorous Formalization of Mathematics 3
643: Constrained Subsets of N, #1 11/3/15 11:57PM
644: Fixed Point Selectors 1 11/16/15 8:38AM
645: Fixed Point Minimizers #1 11/22/15 7:46PM
646: Philosophy of Incompleteness 1 Nov 24 17:19:46 EST 2015
647: General Incompleteness almost everywhere 1 11/30/15 6:52PM
648: Necessary Irrelevance 1 12/21/15 4:01AM
649: Necessary Irrelevance 2 12/21/15 8:53PM
650: Necessary Irrelevance 3 12/24/15 2:42AM
651: Pi01 Incompleteness Update 2/2/16 7:58AM
652: Pi01 Incompleteness Update/2 2/7/16 10:06PM
653: Pi01 Incompleteness/SRP,HUGE 2/8/16 3:20PM
654: Theory Inspired by Automated Proving 1 2/11/16 2:55AM
655: Pi01 Incompleteness/SRP,HUGE/2 2/12/16 11:40PM
656: Pi01 Incompleteness/SRP,HUGE/3 2/13/16 1:21PM
657: Definitional Complexity Theory 1 2/15/16 12:39AM
658: Definitional Complexity Theory 2 2/15/16 5:28AM
659: Pi01 Incompleteness/SRP,HUGE/4 2/22/16 4:26PM
660: Pi01 Incompleteness/SRP,HUGE/5 2/22/16 11:57PM
661: Pi01 Incompleteness/SRP,HUGE/6 2/24/16 1:12PM
662: Pi01 Incompleteness/SRP,HUGE/7 2/25/16 1:04AM
663: Pi01 Incompleteness/SRP,HUGE/8 2/25/16 3:59PM
664: Unsolvability in Number Theory 3/1/16 8:04AM
665: Pi01 Incompleteness/SRP,HUGE/9 3/1/16 9:07PM
666: Pi01 Incompleteness/SRP,HUGE/10 13/18/16 10:43AM
667: Pi01 Incompleteness/SRP,HUGE/11 3/24/16 9:56PM
668: Pi01 Incompleteness/SRP,HUGE/12 4/7/16 6:33PM
669: Pi01 Incompleteness/SRP,HUGE/13 4/17/16 2:51PM
670: Pi01 Incompleteness/SRP,HUGE/14 4/28/16 1:40AM
671: Pi01 Incompleteness/SRP,HUGE/15 4/30/16 12:03AM
672: Refuting the Continuum Hypothesis? 5/1/16 1:11AM
673: Pi01 Incompleteness/SRP,HUGE/16 5/1/16 11:27PM
674: Refuting the Continuum Hypothesis?/2 5/4/16 2:36AM
675: Embedded Maximality and Pi01 Incompleteness/1 5/7/16 12:45AM
676: Refuting the Continuum Hypothesis?/3 5/10/16 3:30AM
677: Embedded Maximality and Pi01 Incompleteness/2 5/17/16 7:50PM
678: Symmetric Optimality and Pi01 Incompleteness/1 5/19/16 1:22AM
679: Symmetric Maximality and Pi01 Incompleteness/1 5/23/16 9:21PM
680: Large Cardinals and Continuations/1 5/29/16 10:58PM
681: Large Cardinals and Continuations/2 6/1/16 4:01AM
682: Large Cardinals and Continuations/3 6/2/16 8:05AM
683: Large Cardinals and Continuations/4 6/2/16 11:21PM
684: Large Cardinals and Continuations/5 6/3/16 3:56AM
685: Large Cardinals and Continuations/6 6/4/16 8:39PM
686: Refuting the Continuum Hypothesis?/4 6/616 9:29PM
687: Large Cardinals and Continuations/7 6/7/16 10:28PM
688: Large Cardinals and Continuations/8 Jun 9 23:41:05 EDT 2016
689: Large Cardinals and Continuations/9 Jun 11 14:51:56 EDT 2016
690: Two Brief Sketches 6/13/16 1:18AM
Harvey Friedman
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