[FOM] 519: New Invariant Maximality

Harvey Friedman hmflogic at gmail.com
Wed Jan 8 22:59:16 EST 2014

 I have a new paper on Concrete Mathematical Incompleteness that focuses on
polished Pi01 incompleteness in the region of SRP and HUGE. There is quite
a bit of extra material, and in this paper, submitted for publication, the
proofs are restricted to proofs using the SRP hierarchy of the simplest
statements of the form "every order invariant relation (graph) has an R
invariant root (clique)", for various natural relations R. Also there is a
new level of explicitly Pi01 form. There is a reasonably polished form of
the design of actual computer investigations that can confirm or refute the
consistency of large cardinals.

I am in a forced migration of my website, and so this pdf file is not going
to be on my website for a week or so. Impatient parties are welcome to
email me privately for a pdf version.

Harvey M. Friedman

Harvey M. Friedman*
Distinguished University Professor of Mathematics, Philosophy, and Computer
Science, Emeritus
Ohio State University
Columbus, Ohio 43235

January 5, 2014

*This research was partially supported by an Ohio State University
Presidential Research Grant and by the John Templeton Foundation grant ID
#36297. The opinions expressed here are those of the author and do not
necessarily reflect the views of the John Templeton Foundation.

Abstract. We present new examples of discrete mathematical statements that
can be proved from large cardinal hypotheses but not within the usual ZFC
axioms for mathematics (assuming ZFC is consistent). These new statements
are provably equivalent to P01 sentences (purely universal statements,
logically analogous to Fermat's Last Theorem) - in particular provably
equivalent to the consistency of strong set theories, including one that is
in explicitly P01 form. The examples live in the rational numbers, with
only order, where the nonnegative integers are distinguished elements. The
statements take the general form: every order invariant W contained in Q^2k
has a maximal S contained in Q^k such that S^2 contained in W, with an
invariance condition. Certain statements of this form are shown to be
provably equivalent to the widely believed Con(SRP), and hence unprovable
in ZFC (assuming ZFC is consistent). Modifications are made, involving a
simple cross section condition, which propels the statement beyond the huge
cardinal hierarchy, to attain equivalence with Con(HUGE). We also present
some nondeterministic constructions of infinite and finite length with some
of the same metamathematical properties. These lead to practical computer
investigations designed to provide arguable confirmation of Con(ZFC) and


1. Introduction
2. Preliminaries.
     2.1. Invariance.
     2.2. Order invariance.
     2.3. N/order invariance.
     2.4. Restricted shifts.
     2.5. (Q,N,+,<).
     2.6. Graphs.
3. Invariant maximal roots and cliques.
     3.1. Maximal roots and cliques in Q^k.
     3.2. Maximal roots and cliques in J^k.
     3.3. Step maximal roots and cliques in Q^k.
     3.4. Inductively maximal cliques in Q^k.
4. N tail equivalence is maximum.
5. Finite inductively sequential cliques.
6. Inductively maximal cliques in Q^<=k.
7. Templates.
8. Computer investigations.
     8.1. Infinite and finite length constructions.
     8.2. Computational aspects.
9. Proofs of invariant maximality.
     9.1. The stationary Ramsey property.
     9.2. Invariant Step Maximal Roots (Nteq).
10. Appendix - formal systems used.
11. References.


I use http://www.math.ohio-state.edu/~friedman/ for downloadable
manuscripts. This is the 519th 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

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
516: Embedded Maximal Cliques/restatement  5/21/13  1:31PM
517: New Concrete Mathematical Incompleteness  8/2/13  9:57PM
518: Polynomial Independence  8/7/13  6:04AM

Harvey M. Friedman
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