[FOM] 516: Embedded Maximal Cliques/restatement

Tue May 21 13:13:31 EDT 2013

THIS RESEARCH WAS PARTIALLY SUPPORTED BY THE JOHN TEMPLETON FOUNDATION

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THIS POSTING IS ENTIRELY SELF CONTAINED

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I have put Embedded Maximal Cliques and Incompleteness, Extended
Abstract, May 20, 2013, 18 pages, on my website at

http://www.math.osu.edu/~friedman.8/manuscripts.html
1. Preprints, Drafts, Manuscripts
manuscript number 76.

There has been significant improvements of various kinds, both
expositional and mathematical. For the general formulations, we now
use order theoretic partial f:J into J that are strictly increasing
and continuous, for the embeddings of maximal cliques in order
invariant graphs on J^k. Here J is a rational interval.

We have greatly improved the explicitly finite and explicitly Pi01
statements, and also the discussion of computer investigations.

FROM THE MANUSCRIPT:

Abstract. We investigate "every order invariant graph on every
Cartesian power of J has an f embedded maximal clique", where J be a
rational interval and f:J into J is partial. For f with finite domain, we
give an elegant necessary and sufficient condition. For general f, it
is necessary that f be strictly increasing, and the investigation of
sufficient conditions leads to statements with unexpected logical
properties. The statements have the ZFC refutation property - it is
provable from the usual ZFC axioms for mathematics, that if the
statement is false then it is refutable in ZFC. The statements are
demonstrably equivalent to the generally believed Con(SRP), which
asserts the consistency of ZFC augmented by a well studied large
cardinal hypothesis. Thus the statements cannot be proved in ZFC
(assuming ZFC is consistent). This phenomena emerges even for
numerical shift functions f:Q>=0 into Q>=0. We investigate associated
infinite sequential constructions and show that "such infinite
constructions have output" is also demonstrably equivalent to
Con(SRP). We present five finite approximations to these
constructions, within the realm of finite mathematics, with the ZFC
refutation property. These statements are in explicitly Pi02, Pi01, Pi04,
Pi02, Pi02 form. The last four are provably equivalent to Con(SRP).
Iterated exponential bounds on the existential quantifiers put the
last three in explicitly Pi01 form. We conclude with a discussion of
associated computer investigations, leading to the prospect of the
confirmation of Con(SRP) by actual practical computer implementations.

1. Graphs, cliques, embeddings, and tame sets.
2. Embedded Maximal Clique Theorems.
3. Infinite sequential constructions.
4. Finite sequential constructions.
5. Computer investigations.

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I use http://www.math.ohio-state.edu/~friedman/ for downloadable
manuscripts. This is the 516th 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 #1-449 can be found
in the FOM archives at
http://www.cs.nyu.edu/pipermail/fom/2010-December/015186.html

450: Maximal Sets and Large Cardinals II  12/6/10  12:48PM
451: Rational Graphs and Large Cardinals I  12/18/10  10:56PM
452: Rational Graphs and Large Cardinals II  1/9/11  1:36AM
453: Rational Graphs and Large Cardinals III  1/20/11  2:33AM
454: Three Milestones in Incompleteness  2/7/11  12:05AM
455: The Quantifier "most"  2/22/11  4:47PM
456: The Quantifiers "majority/minority"  2/23/11  9:51AM
457: Maximal Cliques and Large Cardinals  5/3/11  3:40AM
458: Sequential Constructions for Large Cardinals  5/5/11  10:37AM
459: Greedy CLique Constructions in the Integers  5/8/11  1:18PM
460: Greedy Clique Constructions Simplified  5/8/11  7:39PM
461: Reflections on Vienna Meeting  5/12/11  10:41AM
462: Improvements/Pi01 Independence  5/14/11  11:53AM
463: Pi01 independence/comprehensive  5/21/11  11:31PM
464: Order Invariant Split Theorem  5/30/11  11:43AM
465: Patterns in Order Invariant Graphs  6/4/11  5:51PM
466: RETURN TO 463/Dominators  6/13/11  12:15AM
467: Comment on Minimal Dominators  6/14/11  11:58AM
468: Maximal Cliques/Incompleteness  7/26/11  4:11PM
469: Invariant Maximality/Incompleteness  11/13/11  11:47AM
470: Invariant Maximal Square Theorem  11/17/11  6:58PM
471: Shift Invariant Maximal Squares/Incompleteness  11/23/11  11:37PM
472. Shift Invariant Maximal Squares/Incompleteness  11/29/11  9:15PM
473: Invariant Maximal Powers/Incompleteness 1  12/7/11  5:13AMs
474: Invariant Maximal Squares  01/12/12  9:46AM
475: Invariant Functions and Incompleteness  1/16/12  5:57PM
476: Maximality, CHoice, and Incompleteness  1/23/12  11:52AM
477: TYPO  1/23/12  4:36PM
478: Maximality, Choice, and Incompleteness  2/2/12  5:45AM
479: Explicitly Pi01 Incompleteness  2/12/12  9:16AM
480: Order Equivalence and Incompleteness
481: Complementation and Incompleteness  2/15/12  8:40AM
482: Maximality, Choice, and Incompleteness 2  2/19/12 7:43AM
483: Invariance in Q[0,n]^k  2/19/12  7:34AM
484: Finite Choice and Incompleteness  2/20/12  6:37AM__
485: Large Large Cardinals  2/26/12  5:55AM
486: Naturalness Issues  3/14/12  2:07PM
487: Invariant Maximality/Naturalness  3/21/12  1:43AM
488: Invariant Maximality Program  3/24/12  12:28AM
489: Invariant Maximality Programs  3/24/12  2:31PM
490: Invariant Maximality Program 2  3/24/12  3:19PM
491: Formal Simplicity  3/25/12  11:50PM
492: Invariant Maximality/conjectures  3/31/12  7:31PM
493: Invariant Maximality/conjectures 2  3/31/12  7:32PM
494: Inv Max Templates/Z+up, upper Z+ equiv  4/5/12  4:17PM
495: Invariant Finite Choice  4/5/12  4:18PM
496: Invariant Finite Choice/restatement  4/8/12  2:18AM
497: Invariant Maximality Restated  5/2/12 2:49AM
498: Embedded Maximal Cliques 1  9/18/12  12:43AM
499. Embedded Maximal Cliques 2  9/19/12  2:50AM
500: Embedded Maximal Cliques 3  9/20/12  10:15PM
501: Embedded Maximal Cliques 4  9/23/12  2:16AM
502: Embedded Maximal Cliques 5  9/26/12  1:21AM
503: Proper Classes of Graphs  10/13/12  12:17PM
504. Embedded Maximal Cliques 6  10/14/12  12:49PM
505: Function Transfer Theory 10/21/12  2:15AM
506: Finite Embedded Weakly Maximal Cliques  10/23/12  12:53AM
507: Finite Embedded Dominators  11/6/12  6:40AM
508: Unique Undefinable Elements  12/22/12  8:08PM
509: A Divine Consistency Proof for Mathematics  12/26/12  2:15AM
510: Unique Undefinable Elements Again  1/9/13  5:07PM
511: A Supernatural Consistency Proof for Mathematics   1/10/13  9:19PM
512: Countable Elementary Extensions  1/11/13  7:31PM
513: Five Supernatural Consistency Proofs for Mathematics  1/14/13  1:13AM
514: Countable Elementary Extensions/again  1/14/13  2:19AM
515: Eight Supernatural Consistency Proofs For Mathematics  1/19/13  2:40PM

Harvey Friedman
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