[FOM] 666: Pi01 Incompleteness/SRP,HUGE/10
Harvey Friedman
hmflogic at gmail.com
Fri Mar 18 10:43:20 EDT 2016
We are in the process of sketching detailed proofs of the claims in
http://www.cs.nyu.edu/pipermail/fom/2016-March/019550.html
starting with the all important Lead Proposition. For the reader's
convenience, we copy the four statements from
http://www.cs.nyu.edu/pipermail/fom/2016-March/019550.html
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.
LEAD FINITE PROPOSITION. Every reflexive order invariant R containedin
N^2k has a fixed point minimizer f sending [n]!^k to some A^k
containedin dom(f) intersect dom(f)-1, without (8k)!-1. We can require
that f live below (8kn)!.
LEAD HUGE PROPOSITION. Every reflexive order invariant R containedin
Q^2k has a <= fixed point minimizer f:A^k into A^k with comparable
nontrivial partial embeddings f_n...n||<n-1.
LEAD FINITE HUGE PROPOSITION. Every reflexive order invariant R
containedin N^2k has a <= fixed point minimizer f sending [n]!^k to
some A^k containedin dom(f), with comparable nontrivial partial
embeddings f_j!...j!||<(j-1)!, 8k < j <= n. We can require that f live
below (8kn)!.
***************************
I have been focusing on the Lead Proposition. The reversal is very
delicate and being hard fought, and there have been some scary
moments. The fight continues. I have a nice new replacement for the
Lead Proposition in case something goes wrong. I am working with
rather small k,n, which I find very satisfying.
In the meantime, I found a very good way of talking about finite
approximations to maximal S^2 containedin V. I now view this new
finite form as the new LEAD FINITE PROPOSITION. This is, like the Lead
Proposition, at the level of SRP.
DEFINITION 1. The height of a rational p, H(p), is the maximum of the
magnitudes of the numerator and denominator in its reduced form
(standard). H(x), The height of x in Q^k, H(x), is
max(H(x_1),...,H(x_k)). The height of finite S containedin Q^k, H(S),
is the maximum of the H(x), x in S. S//r = {x in S: H(x) <= r}.
The notion of height of a rational is standard. I'm not sure that this
notion of height of a rational vector is standard. The results would
be the same if I used, e.g., the height of x as the sum of the heights
of the coordinates of x.
DEFINITION 2. Let V containedin Q^k. S^2 containedin V is k-maximal if
and only if there is no H(x) <= k with (S disjointunion {x})^2
containedin V.
(NEW) LEAD FINITE PROPOSITION. For all order invariant V containedin
Q[0,n]^k//(8kn)!, some k-maximal S^2 containedin V extends to a
#(S)-maximal S'^2 containedin V that is partially embedded by the
function p if p < 1; p+1 if p = 1,...,n-1.
Compare the above with
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.
The above is explicitly P01. It is provably equivalent to Con(SRP) over EFA.
**********************************************************
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 666th 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
662: Pi01 Incompleteness/SRP,HUGE/7 2/25/16 1:04AM
663: Pi01 Incompleteness/SRP,HUGE/8 2/25/16 3:59PM
664: Unsolvability in Number Theory 3/1/16 8:04AM
665: Pi01 Incompleteness/SRP,HUGE/9 3/1/16 9:07PM
Harvey Friedman
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