[FOM] 558: New Pi01/more

[Harvey Friedman]

Here we improve on
http://www.cs.nyu.edu/pipermail/fom/2014-October/018349.html in some
important ways.

In this posting, we will work entirely in N and not use order invariance at all.

In http://www.cs.nyu.edu/pipermail/fom/2014-October/018349.html we had
+1 built into the notion of reduction. We now use reduction in its
purest form without building in +1.

DEFINITION 1. Let R be contained in N^2k = N^k x N^k. x reduces to y
if and only if x R y and max(x) > max(y). S is irreducible in R if and
only if S containedin N^k and no element of S reduces to any element
of S.

DEFINITION 2. Let R containedin N^2k. A_1 red ... red A_r for S if and only if
i. A_1,...,A_r are subsets of N.
ii. S is a subset of N^k.
iii. Every element of  (A_i )^k\S reduces to an element of A_i+1 intersect S.

ZFC is insufficient to handle matters even for r = 4. So we will
concentrate on r = 4 at this point.

PROPOSITION 1. Every R containedin N^2k has an irreducible set for
which some infinite A red B red C red D has B+1 containedin C\A.

NOTE: Officially, we take "infinite" here to apply to A,B,C,D.
However, we can take "infinite" to apply to just A and obtain the same
results.

THEOREM 2. Proposition 1 is provably equivalent to Con(SMAH) over ACA.

We haven't yet seriously tried to reduce the base theory ACA. Here
SMAH is the strongly Mahlo cardinal hierarchy.

TEMPLATE A. Every R containedin N^2k has an irreducible set for which
some infinite A red B red C red D has a given Boolean equation in
A,B,C,D,A+1,B+1,C+1,D+1.

Of course, we want to handle Boolean equations in infinite A_1,...,A_n.

What about explicitly finite versions? Here we use order invariance.

The first step is to simply move to order invariant R.

PROPOSITION 3. Every order invariant R containedin N^2k has an
irreducible set for which some infinite A red B red C red D has B+1
containedin C\A.

Order invariance allows us to be specific about A. We write E! = {n!: n in E}.

PROPOSITION 4. Every order invariant R containedin N^2k has an
irreducible set for which some infinite (8kN)! red B red C red D has
B+1 containedin C\(8kN)!.

Then we can move to the finite rather smoothly.We write [r] = {0,...,r}.

PROPOSITION 5. Every order invariant R containedin N^2k has a finite
irreducible set for which some finite (8k[r])! red B red C red D has
B+1 containedin C\(8k[r])!. We can require the irreducible set and
B,C,D to live <= (8kr)!+1.

THEOREM 7. Propositions 3-5 are provably equivalent to Con(SMAH) over ACA.

************************************************************
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 558th 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

Harvey Friedman