[FOM] 530: Friendlier Perfect Pi01

[Harvey Friedman]

I have upgraded the extended abstract #82
https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
with
material designed to make the perfect Pi01 mathematical incompleteness yet
more friendly for the mathematics community.


ORDER INVARIANT RELATIONS AND
INCOMPLETENESS
by
Harvey M. Friedman*
Distinguished University Professor of Mathematics,
Philosophy, and Computer Science Emeritus
Ohio State University
August 22, 2014

EXTENDED ABSTRACT

*This research was partially supported 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. Every order invariant subset of Q[0,n]^2k has all
sections at strictly increasing r tuples of positive
integers ≤ n agreeing below 1 (trivial). Every order
invariant subset of Q[0,n]^2k has a maximal square whose
sections at strictly increasing r tuples of positive
integers ≤ n agree below 1 (nontrivial). We prove the latter
in ZFC augmented with a standard large cardinal hypothesis,
and show that ZFC does not suffice (assuming ZFC is
consistent). We also establish this for a number of
variants, including an explicitly finite form. We refer to
these developments as
​
Pi01 mathematical incompleteness

Just about the most perfect of the perfect would be something like this:

PROPOSITION. E
​v​
ery order invariant subset of Q^8 has a maximal square whose sections at
(1,2,3) and (2,3,4) agree below 1.
PROPOSITION. Every order invariant graph on Q^4 has a has a maximal clique
whose sections at (1,2,3) and (2,3,4) agree below 1.

I don't know how to prove such statements are independent of ZFC, although
I can prove them using large cardinals.

****************************************
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 530th 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

Harvey Friedman
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