[FOM] 677:Embedded Maximality and Pi01 Incompleteness/2

Tue May 17 19:50:37 EDT 2016

SO HERE IS WHERE WE ARE NOW.

1. I prefer as the lead statement, the version in
http://www.cs.nyu.edu/pipermail/fom/2016-May/019802.html But for some
audiences, particularly graph theorists, the previous versions in
http://www.cs.nyu.edu/pipermail/fom/2016-May/019792.html might be
preferable. So these previous ones will always be presented as
alternative formulations.

2. In the lead statement, we put a nice equivalence relation which we
call SIMILARITY on the subsets of Q^k|<=n. It says that the subsets of
cardinality <=2 from the two sets, are the same up to order
isomorphism. For each k, there are only finitely many equivalence
classes. The lead statement says that: in each equivalence class of
subsets of Q^k|<=n under similarity, some maximal element is partially
embedded by the function p if p < 0; p+1 if p = 0,...,n-1. This is
implicitly Pi01 via Goedel's Completeness Theorem, and is provably
equivalent to Con(SRP) over WKL_0.

3. In the previous statements, we use order invariant subsets of
Q^k|<=n, maximal squares, and the same self embedding condition. Also
the graph theoretic formulation uses order invariant graphs on
Q^k|<=n, maximal cliques, and the same self embedding condition. The
formulation in 2 is more direct. But both need to be mentioned.

In this posting I want to focus on the explicitly Pi01 statements
provably equivalent to Con(SRP) over EFA. I like the present
statements a lot better than what I did in
http://www.cs.nyu.edu/pipermail/fom/2016-May/019802.html.

DEFINITION. [t] = {1,...,t}. A,B containedin Q^k are 3-similar if and
only if they have the same <= 3 element subsets up to order
isomorphism. In the family of sets K, S is maximal over X if and only
if S in K has no proper superset in K intersect POW(S union X).

Partial self embeddings have been defined previously.

FINITE EMBEDDED MAXIMALITY. FEM. In every equivalence class of subsets
of [(8k)!!k]^k under 3-similarity, some S, S intersect E^k are
respectively maximal over E^k, (8k)!!N^k, and partially self embedded
by the function m if m in [(8k)!!-1]; m+(8k)!! if m in (8k)!![k-1].

Obviously, FEM is in explicitly Pi01 form.

THEOREM. FEM is provably equivalent to Con(SRP) over EFA.

***********************************************
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 677th 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

Harvey Friedman