[FOM] 662: Pi01 Incompleteness/SRP,HUGE/7

Thu Feb 25 01:04:18 EST 2016

I think I finally have my Lead Finite Proposition. The reason I can
move fast with this is because I have, maybe, a dozen or so ideas for
the Lead Finite Proposition that only reach a certain level of
simplicity and naturalness, but these ideas can be combined in various
ways. Generally, about 3 ideas get combined in each attempt. So that's
a lot of combinations to sort through, together with detailed working
knowledge of how to obtain models of large cardinals from the various
combinations.

For the Lead Finite Proposition, it appears - at least right now -
that the best way of going about this is to use the serious power of
the fixed point minimizers I have been talking about. Also to move
from Q into N, and avoid using embeddings. Since finite fixed point
minimizers can be trivial (the empty function), we are going to have
to use a clause that is not as strikingly simple as the clause in the
Lead Proposition (very stable at this point)

LEAD PROPOSITION. For all order invariant V containedin Q[0,n]^k, some
maximal S^2 containedin V is partially embedded by the function p if p
< 1; p+1 if p = 1,...,n-1.

which simply asserts maximality. Because I am only asserting
maximality, I can afford, in the Lead Proposition, to have this
displayed embedding as the side condition.

So it seems like I am forced to avoid such a side condition for the
Lead Finite Proposition if it is going to have any chance of crossing
that magic line into the Perfectly Natural.

Just the right combination of my previous ideas seem to do the trick here.

LEAD FINITE PROPOSITION

Recall p stands for rationals, and i,k,n stand are positive integers.

DEFINITION 1. [p] = {i: i <= p}. For A containedin N, A! = {n!: n in
A}. Let f:A^k into B^k and C,D containedin A. f sends C into D if and
only if f[C^k] containedin D^k.

DEFINITION 2. Let R containedin [n]^k. A fixed point minimizer is a
function f:A^k into [n]^k, A containedin [n], where every f(x) is the
lexicographically least y = f(y) with R(x,y).

LEAD FINITE PROPOSITION. Every reflexive order invariant R containedin
[n!]^2k has a fixed point minimizer f:A^k into [n!]^k sending [n/8k]!
containedin A into A intersect A-1  \ {(8k)!-1}.

Obviously Lead Finite Proposition is explicitly Pi01.

THEOREM 1. Lead Finite Proposition is provable equivalent to Con(MAH) over EFA.

Recall MAH is ZFC + {there exists an n-Mahlo cardinal}_n.

Assuming that this remains the Lead Finite Proposition, it suggests a
number of very interesting weakenings to be investigated. Here are
some:

1. Modifying. the punch line A intersect A-1 \ {(8k)!-1}.
1a. Removing {(8k)-1}!. Certainly provable in PRA, but EFA might
require some thought.
1b. Removing A-1. Should be provable in EFA.

2. Keeping the punch line A intersect A-1 \ {(8k)!-1}. Changing n!.
The interesting thing is to make n a function of k. The higher the
function of k that n is, the stronger the statement.
2a. Replace n! with k!!. Should be about Z_2.
2b. Replace n! with k!...!. r factorials, r outside quantifier Should
be about Z_r, where there are r factorials.
2c. Replace n! with k!...!, where there are k !'s. Should be around ZC.
2d. Ackerman function of k. Should be around ZFC.
2e. <epsilon_0 recursive functions of k. Should climb through the
Mahlo cardinal hierarchy.
2f. Probably one has some delicate low level stuff going through
fragments of PA.

But first let's see how long the Lead Finite Proposition lasts. As
usual, I think it will last forever (smile).

**********************************************************
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 662nd 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 #1-599 can be found at the FOM posting
http://www.cs.nyu.edu/pipermail/fom/2015-August/018887.html

600: Removing Deep Pathology 1  8/15/15  10:37PM
601: Finite Emulation Theory 1/perfect?  8/22/15  1:17AM
602: Removing Deep Pathology 2  8/23/15  6:35PM
603: Removing Deep Pathology 3  8/25/15  10:24AM
604: Finite Emulation Theory 2  8/26/15  2:54PM
605: Integer and Real Functions  8/27/15  1:50PM
606: Simple Theory of Types  8/29/15  6:30PM
607: Hindman's Theorem  8/30/15  3:58PM
608: Integer and Real Functions 2  9/1/15  6:40AM
609. Finite Continuation Theory 17  9/315  1:17PM
610: Function Continuation Theory 1  9/4/15  3:40PM
611: Function Emulation/Continuation Theory 2  9/8/15  12:58AM
612: Binary Operation Emulation and Continuation 1  9/7/15  4:35PM
613: Optimal Function Theory 1  9/13/15  11:30AM
614: Adventures in Formalization 1  9/14/15  1:43PM
615: Adventures in Formalization 2  9/14/15  1:44PM
616: Adventures in Formalization 3  9/14/15  1:45PM
617: Removing Connectives 1  9/115/15  7:47AM
618: Adventures in Formalization 4  9/15/15  3:07PM
619: Nonstandardism 1  9/17/15  9:57AM
620: Nonstandardism 2  9/18/15  2:12AM
621: Adventures in Formalization  5  9/18/15  12:54PM
622: Adventures in Formalization 6  9/29/15  3:33AM
623: Optimal Function Theory 2  9/22/15  12:02AM
624: Optimal Function Theory 3  9/22/15  11:18AM
625: Optimal Function Theory 4  9/23/15  10:16PM
626: Optimal Function Theory 5  9/2515  10:26PM
627: Optimal Function Theory 6  9/29/15  2:21AM
628: Optimal Function Theory 7  10/2/15  6:23PM
629: Boolean Algebra/Simplicity  10/3/15  9:41AM
630: Optimal Function Theory 8  10/3/15  6PM
631: Order Theoretic Optimization 1  10/1215  12:16AM
632: Rigorous Formalization of Mathematics 1  10/13/15  8:12PM
633: Constrained Function Theory 1  10/18/15 1AM
634: Fixed Point Minimization 1  10/20/15  11:47PM
635: Fixed Point Minimization 2  10/21/15  11:52PM
636: Fixed Point Minimization 3  10/22/15  5:49PM
637: Progress in Pi01 Incompleteness 1  10/25/15  8:45PM
638: Rigorous Formalization of Mathematics 2  10/25/15 10:47PM
639: Progress in Pi01 Incompleteness 2  10/27/15  10:38PM
640: Progress in Pi01 Incompleteness 3  10/30/15  2:30PM
641: Progress in Pi01 Incompleteness 4  10/31/15  8:12PM
642: Rigorous Formalization of Mathematics 3
643: Constrained Subsets of N, #1  11/3/15  11:57PM
644: Fixed Point Selectors 1  11/16/15  8:38AM
645: Fixed Point Minimizers #1  11/22/15  7:46PM
646: Philosophy of Incompleteness 1  Nov 24 17:19:46 EST 2015
647: General Incompleteness almost everywhere 1  11/30/15  6:52PM
648: Necessary Irrelevance 1  12/21/15  4:01AM
649: Necessary Irrelevance 2  12/21/15  8:53PM
650: Necessary Irrelevance 3  12/24/15  2:42AM
651: Pi01 Incompleteness Update  2/2/16  7:58AM
652: Pi01 Incompleteness Update/2  2/7/16  10:06PM
653: Pi01 Incompleteness/SRP,HUGE  2/8/16  3:20PM
654: Theory Inspired by Automated Proving 1  2/11/16  2:55AM
655: Pi01 Incompleteness/SRP,HUGE/2  2/12/16  11:40PM
656: Pi01 Incompleteness/SRP,HUGE/3  2/13/16  1:21PM
657: Definitional Complexity Theory 1  2/15/16  12:39AM
658: Definitional Complexity Theory 2  2/15/16  5:28AM
659: Pi01 Incompleteness/SRP,HUGE/4  2/22/16  4:26PM
660: Pi01 Incompleteness/SRP,HUGE/5  2/22/16  11:57PM
661: Pi01 Incompleteness/SRP,HUGE/6  2/24/16  1:12PM

Harvey Friedman