[FOM] 579: Impossible Counting

Tue May 26 20:58:18 EDT 2015

In posting 566
http://www.cs.nyu.edu/pipermail/fom/2015-January/018517.html  I wrote:

COMPLEXITY DEFINITION. Let M be a relational structure. We use and,
or, not, if then, iff, formal, therexists, =, together with the
relation, constant, and function symbols interpreted by M. The
complexity of a formula in L(M) is the total number of relation,
constant, and function symbols appearing in the formula.

To clarify this, I am talking about the total number of *occurrences*
of relation, constant, and function symbols appearing in the formula.

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My Concrete Mathematical Incompleteness project focuses on identifying
specific single Perfectly Mathematically Natural concrete statements
not decided in ZFC and beyond. There are also the focus on identifying
specific Perfectly Mathematically Natural Templates of concrete
statements, which can and can only be decided by going well beyond
ZFC.

There is a somewhat different aim which, at the moment, uses separate
methods which are delicate, but not comparable in depth to those used
in the aforementioned projects.

Specifically, identifying specific Perfectly Mathematically Natural
Concrete Count Problems which cannot be solved in ZFC and beyond. Here
we are not expecting these Count Problems to be solved using large
cardinals. In fact, the Count Problems are absolutely undecidable in
the sense that no consistent adequate formal system of reasonable size
is sufficient to solve the Count Problem.

Of course, in both programs, if one deletes the Perfectly
Mathematically Natural condition, then it is well known by Goedel and
standard elaborations on Goedel/Turing.

What is a Count Problem?

A finite mathematical object is specified, and we ask for an exact
count on its cardinality.

HOW MANY BINARY OPERATIONS ARE THERE UP TO 12-SIMILARITY? - i.e., they
have the same restrictions to 12 element sets up to isomorphism.

ZFC, even ZFC augmented with standard large cardinal hypotheses, is
insufficient to get an exact count.

In a sense, this places an upper bound on the complexity of
mathematics, or the complexity of mathematical thought. I am not clear
at this point just how to say this kind of thing.

There is a draft with some proofs at

https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
#88. Impossible Counting, May 26, 2015, 9 pages, draft.

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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 579th 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 #1-527 can be found at the FOM posting
http://www.cs.nyu.edu/pipermail/fom/2014-August/018092.html

528: More Perfect Pi01  8/16/14  5:19AM
529: Yet more Perfect Pi01 8/18/14  5:50AM
530: Friendlier Perfect Pi01
531: General Theory/Perfect Pi01  8/22/14  5:16PM
532: More General Theory/Perfect Pi01  8/23/14  7:32AM
533: Progress - General Theory/Perfect Pi01 8/25/14  1:17AM
534: Perfect Explicitly Pi01  8/27/14  10:40AM
535: Updated Perfect Explicitly Pi01  8/30/14  2:39PM
536: Pi01 Progress  9/1/14 11:31AM
537: Pi01/Flat Pics/Testing  9/6/14  12:49AM
538: Progress Pi01 9/6/14  11:31PM
539: Absolute Perfect Naturalness 9/7/14  9:00PM
540: SRM/Comparability  9/8/14  12:03AM
541: Master Templates  9/9/14  12:41AM
542: Templates/LC shadow  9/10/14  12:44AM
543: New Explicitly Pi01  9/10/14  11:17PM
544: Initial Maximality/HUGE  9/12/14  8:07PM
545: Set Theoretic Consistency/SRM/SRP  9/14/14  10:06PM
546: New Pi01/solving CH  9/26/14  12:05AM
547: Conservative Growth - Triples  9/29/14  11:34PM
548: New Explicitly Pi01  10/4/14  8:45PM
549: Conservative Growth - beyond triples  10/6/14  1:31AM
550: Foundational Methodology 1/Maximality  10/17/14  5:43AM
551: Foundational Methodology 2/Maximality  10/19/14 3:06AM
552: Foundational Methodology 3/Maximality  10/21/14 9:59AM
553: Foundational Methodology 4/Maximality  10/21/14 11:57AM
554: Foundational Methodology 5/Maximality  10/26/14 3:17AM
555: Foundational Methodology 6/Maximality  10/29/14 12:32PM
556: Flat Foundations 1  10/29/14  4:07PM
557: New Pi01  10/30/14  2:05PM
558: New Pi01/more  10/31/14 10:01PM
559: Foundational Methodology 7/Maximality  11/214  10:35PM
560: New Pi01/better  11/314  7:45PM
561: New Pi01/HUGE  11/5/14  3:34PM
562: Perfectly Natural Review #1  11/19/14  7:40PM
563: Perfectly Natural Review #2  11/22/14  4:56PM
564: Perfectly Natural Review #3  11/24/14  1:19AM
565: Perfectly Natural Review #4  12/25/14  6:29PM
566: Bridge/Chess/Ultrafinitism 12/25/14  10:46AM
567: Counting Equivalence Classes  1/2/15  10:38AM
568: Counting Equivalence Classes #2  1/5/15  5:06AM
569: Finite Integer Sums and Incompleteness  1/515  8:04PM
570: Philosophy of Incompleteness 1  1/8/15 2:58AM
571: Philosophy of Incompleteness 2  1/8/15  11:30AM
572: Philosophy of Incompleteness 3  1/12/15  6:29PM
573: Philosophy of Incompleteness 4  1/17/15  1:44PM
574: Characterization Theory 1  1/17/15  1:44AM
575: Finite Games and Incompleteness  1/23/15  10:42AM
576: Game Correction/Simplicity Theory  1/27/15  10:39 AM
577: New Pi01 Incompleteness  3/7/15  2:54PM
578: Provably Falsifiable Propositions  3/7/15  2:54PM

Harvey Friedman