[FOM] 502: Embedded Maximal Cliques 5

[Harvey Friedman]

THIS RESEARCH WAS PARTIALLY SUPPORTED BY THE JOHN TEMPLETON FOUNDATION

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We have revamped sections 9 and 11. We have made some other minor
upgrades and added polish.

We have posted the revised September 26, 2012 version of the entire
Extended Abstract at my website at

http://www.math.osu.edu/~friedman.8/
downloadable manuscripts
manuscript 72, Embedded Maximal Cliques and Incompleteness, September
26, 2012, 17 pages.

This supersedes the September 23, 2012 version that was previously
posted on my website.

For the reader's convenience, we repeat the Abstract for the Extended
Abstract here, and the Table of Contents (again changed a little):

Abstract. Every order invariant graph on Q≥0^k has an f embedded
maximal clique, where f is +1 on {0,...,n}. Every order invariant
graph on Q≥0^k has an f embedded maximal clique, where f is +1 on
{0,...,n} extended by the identity on Q>n+1. We give a proof of the
first statement within the usual ZFC axioms for mathematics. We give a
proof of the second statement (and more general statements) that goes
well beyond ZFC, and establish that ZFC does not suffice (assuming ZFC
is consistent). As a consequence of the Gödel completeness theorem,
both statements are Π01. We present a nondeterministic algorithm for
generating finite f embedded cliques. We prove the explicitly Π01
sentence asserting that the algorithm succeeds, by going well beyond
ZFC - and show that ZFC does not suffice (assuming ZFC is consistent).
We also propose a practical form of the algorithm, with the
expectation that actual certificates will be produced. We suggest that
this can be used as a confirmation of the consistency, or pragmatic
consistency, of ZFC and some of its far reaching extensions.

1. Graphs, cliques, embeddings.
2. Order invariant and order theoretic.
3. OIG(J,f).
4. OIG(J,f1,...,fn).
5. OIG characterizations
6. Total embeddings.
7. n-invariance.
8. General conjectures.
9. Finite embedded cliques. .
10. Extremely strong statement.
11. Certificates.

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I use http://www.math.ohio-state.edu/~friedman/ for downloadable
manuscripts. This is the 502nd 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

Harvey Friedman