[FOM] 660: Pi01 Incompleteness/SRP,HUGE/5
Harvey Friedman
hmflogic at gmail.com
Mon Feb 22 23:12:57 EST 2016
In http://www.cs.nyu.edu/pipermail/fom/2016-February/019529.html the
items in section 4 were misnumbered using 3 instead of 4. This is
corrected below.
The Lead Proposition is particularly stable these days. The Lead
Finite Proposition also looks stable, although far newer.
But Lead Finite Proposition' should be removed, and Propositions 3.2,
4.1 need to be corrected. These errors were based on my temporary
carelessness with the notion of partial embedding being used. Also
some 0's should be 1's (in some places, it doesn't make any
difference). Here is the corrected list.
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. Corresponds to SRP.
LEAD FINITE PROPOSITION. For all order invariant V containedin
Q[0,n]^kr, some finitely maximal S^r containedin V is partially
embedded by the function p if p < 1; p+1 if p = 1,...,n-1. Corresponds to SRP.
PROPOSITION 3.2. Every reflexive order invariant R containedin Q^2k
has a fixed point minimizer f:A^k into A^k containing N^k, partially
embedded by the function p if p < 1; p+1 if p = 1,...,n-1. Corresponds to SRP.
PROPOSITION 3.4. Every reflexive order invariant R containedin Q^2k+2
has a <= fixed point minimizer f:A^k+1 into A^k+1 containing N^k
nontrivially embedded by the function union_n f_n...n|<n-1. Corresponds to HUGE.
PROPOSITION 4.1. Every reflexive order invariant R containedin Q^2k
has an n-fixed point minimizer partially embedded by the function p if
p < 1; p+1 if p = 1,...,n-1. Corresponds to SRP.
PROPOSITION 4.2. Every reflexive order invariant R containedin Q^2k+2
has a n,<= fixed point minimizer partially embedded by the function
f_n...n|<n-1, mapping 1,...,n-2 to 2,...,n-1. Corresponds to HUGE.
Of the many definitions of partial embeddings and embeddings, the one
that we are using is as follows.
EMBEDDING DEFINITIONS. Let S containedin Q^k. A partial embedding of S
is a partial function h:Q into Q such that for all x in dom(h)^k, x in
S iff hx in S. Here h acts coordinatewise. An embedding of f:A^k into
A^k containedin Q^k is a partial embedding of (the graph of) f with
domain A. h is nontrivial if and only if h is the not the identity
function on its domain.
Of many other reasonable definitions of partial embeddings and
embeddings, there is a weakening which we call a partial lifting and
lifting.
LIFTING DEFINITIONS. Let S containedin Q^k. A partial lifting of S is
a partial function h:Q into Q such that for all x in dom(h)^k, x in S
implies hx in S. Here h acts coordinatewise. An lifting of f:A^k into
A^k containedin Q^k is a partial lifting of (the graph of) f with
domain A. h is nontrivial if and only if h is the not the identity
function on its domain.
Using Liftings, we have the following lists of Propositions. It is
only the fixed point minimizer statements that can use the weaker
Liftings.
PROPOSITION A. Every reflexive order invariant R containedin Q^2k
has a fixed point minimizer f:A^k into A^k containing N^k, partially
lifted by the function p if p < 1; p+1 if p = 1,...,n-1. Corresponds to SRP.
PROPOSITION A'. Every reflexive order invariant R containedin Q^2k
has a fixed point minimizer f:A^k into A^k containing N^k, partially
lifted by the function p if p < 1; p+1 if 1 <= p <= n-1. Corresponds to SRP.
PROPOSITION B. Every reflexive order invariant R containedin Q^2k+2
has a <= fixed point minimizer f:A^k+1 into A^k+1 containing N^k
nontrivially lifted by the function union_n f_n...n|<n-1. Corresponds to HUGE.
PROPOSITION C. Every reflexive order invariant R containedin Q^2k
has an n-fixed point minimizer partially lifted by the function p if
p < 1; p+1 if p = 1,...,n-1. Corresponds to SRP.
PROPOSITION C'. Every reflexive order invariant R containedin Q^2k
has an n-fixed point minimizer partially lifted by the function p if
p < 1; p+1 if 1 <= p <= n-1. Corresponds to SRP.
PROPOSITION D. Every reflexive order invariant R containedin Q^2k+2
has a <= n-fixed point minimizer partially lifted by the function
f_n...n|<n-1, mapping 1,...,n-2 to 2,...,n-1. Corresponds to HUGE.
We can move the finite statements above from Q to Z, with adjustments.
However, it is not yet clear just how much there is to gain by doing
this.
**********************************************************
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 660th 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
Harvey Friedman
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