[FOM] 687:Large Cardinals and Continuations/7

Tue Jun 7 22:28:42 EDT 2016

We now want to focus on the SYMMETRY being used in Continuation Theory.

All of the statements assert that every seed can blossom into a fully
grown plant with a certain "symmetry" property.

For the basic statements, the "symmetry" properties used are all the
same. Namely, that the following function forms an embedding:

p if p < 0; p+1 if p = 0,...,n-1.

This is really a COMBINATORIAL EMBEDDING condition, of course. It does
represent a kind of symmetry.

But now we want to put this symmetry into the more CONVENTIONAL
SYMMETRY contexts.

First, let us recall the main statement to which we are applying symmetry.

EMBEDDED MAXIMAL CONTINUATIONS. EMC. For finite subsets of Q^k|<0,
some maximal continuation within Q^k|<=n is partially self embedded by
the function p if p < 0; p+1 if p = 0,...,n-1.

What is really going on, in terms of conventional symmetry, is

SYMMETRIC MAXIMAL CONTINUATIONS. SMC. For finite subsets of Q^2k|<0,
some maximal continuation within Q^2k|<=k is +(0*k,1*k) invariant on
Q^k|<0 x {(0,...,k-1)}.

So here are the relevant definitions.

DEFINITION 1. p*k is (p,...,p) in Q^k. Let S,C,{x} containedin Q^k . S
is +x invariant on C if and only if (S intersect C) + x = (S + x
intersect C + x).

DEFINITION 2. A rational hyperrectangle is a product of finitely many
intervals in Q with endpoints from Q union {-infinity,infinity}.

This whole development suggests the following template.

HYPERRECTANGLE CONTINUATION TEMPLATE. HCT. Let A,B,C be rational
hyperrectangles and x be a rational vector, all in the same dimension.
For finite subsets of A, some
maximal continuation within B is +x invariant on C.

Obviously SMC is a special case of HCT, and is equivalent to Con(SRP)
over WKL_0.

CONJECTURE. Every instance of HCT is refutable in RCA_0 or provable iin SRP.

We have a limited understanding of HCT. The case where C is a
product of singletons and intervals unbounded in B, seems to be doable
by present methods. Here unbounded in B means that one or both
endpoints are endpoints of B.

This development also makes perfectly good sense for the explicitly
Pi01 form of SMC. So we get FSMC:

FINITE SYMMETRIC MAXIMAL CONTINUATIONS. FSMC. For finite subsets of
Q^2k|<0, some two successive finite rich continuations within Q^2k|<=k
are +(0*k,1*k) invariant on Q^k|<0 x {(0,...,k-1)}.

The above is provably equivalent to Con(SRP) over EFA.

FINITE HYPERRECTANGLE CONTINUATION TEMPLATE. FHCT. Let A,B,C be
rational hyperrectangles and x be a rational vector, all in the same
dimension. For finite subsets of A,
some two successive finite rich continuations within B are +x invariant on C.

CONJECTURE. Every instance of FHCT is refutable in EFA or provable in SRP.

In the context of smoothly maximal continuations, we have the same
development. In this context, we can prove the above two conjectures.

***********************************************
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 687th 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

Harvey Friedman