[FOM] 548: New Explicitly Pi01

[Harvey Friedman]

*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.

I have put a new version of Universal Properties and Incompleteness,
dated October 4, 2014, at
https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
#85. The main improvement concerns new explicitly Pi01 statements
corresponding to SRP and to HUGE. Here is the new one corresponding to
SRP:

DEFINITION. Q<m> = {i/j in Q[-m,m]: |i|,|j| <= m!}. R containedin
Q<m>^n is order invariant if and only if for all order equivalent x,y
in Q<m>^n, x in R iff y in R. N is the set of all nonnegative
integers. N! = {1!,2!,...}.

DEFINITION. Let R containedin Q<m>^2n and S containedin Q<m>^n. S is R
independent if and only if S containedin Q<m>^n, and no two distinct
elements of S are related by R. B red(R,S) C red(R,S) D if and only if
for all x in B^n (C^n) intersect Q<m>^n, there exists y in C^n (D^n)
such that x R y and max(x) >= max(y). B red(R,S) C red(R,S) D, with or
without p if and only if B red(R,S) C red(R,S) D, and B\{p} red(R,S)
C\{p} red(R,S) D\{p}.  The latter is read "B R-reduces by S to C,
which R-reduces to by S to D, with or without p".

THEOREM. (N! to Z to Q). Every order invariant R contained in Q<m>^2n
has an independent S such that N! red(R,S) Z red(R,S) Q.

PROPOSITION. (N! to Z to Q). Every order invariant R contained in
Q<m>^2n has an independent S such that N! red(R,S) Z red(R,S) Q, with
or without (8n)!-1.

THEOREM. The above Proposition is provably equivalent to Con(SMAH)
over ACA. (SMAH = strongly Mahlo cardinal hierarchy). It is provable
in SMAH+ but not in any consistent SMAH[k]. In particular, it is not
provable in ZFC (assuming ZFC is consistent).

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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 548th 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

Harvey Friedman