[FOM] 785: Revolutionary Possibilities/4
Harvey Friedman
hmflogic at gmail.com
Mon Jan 22 00:38:10 EST 2018
Let's review my previously favorite that is proved using 1-Con(PA).
https://cs.nyu.edu/pipermail/fom/2018-January/020824.html
THEOREM 1. For all k >= 1 there exists n >= 1 whose k maximal factors
have increasing (<=) quesidues.
**********************
We now want to use a different kind of quadratic residue.
DEFINITION. The upper quesidue is the residue of n^2 mod the least
prime greater than |n|. The lower quesidue is the residue of n^2 mod
the least prime greater than some >1 factor of n. The quesidue of
-1,0,1 is undefined.
The quesidues in
https://cs.nyu.edu/pipermail/fom/2018-January/020824.html are now
called the UPPER quesidues.
We now want to work with lower quesidues.
MY CURRENT FAVORITE AT THE PA LEVEL
BASED ON LOWER QUESIDUES
THEOREM 1. For all positive k there exists positive n whose at least k
positive factors have a total of at most (logk)(1+logk)/2 lower
quesidues.
Here c(x) is the ceiling of x.
We prove Theorem 1 using more than PA, in fact we use 1-Con(PA).
Theorem 1 rests on only a basic obvious property of quesidues:
THEOREM 2. Let f:Z+ into Z+ take on only finitely many values at the
positive multiples of any given t. For all positive k there exists
positive n such that f assumes at most (logk)(1+logk)/2 values at the
at least k factors of n greater than 1.
NOTE: You can use r-th power residues instead of quadratic residues
here, in case this isn't "hard" enough.
So the philosophy of Theorem 1 is that
1. We know it holds because it holds of any suitable f (Theorem 2).
2. The particular f in Theorem 1 is apparently "number theoretically
random" in the informal sense that number theorists use.
3. If, as does happen, the number theorists do manage to prove some
basic statistical facts about the lower quesidue function then that is
not going to be suitably relevant to prove Theorem 1.
4. Consequently no such exact information is going to be rigorously
proved by current number theorists.
The weak part of this line of reasoning is of course 4. So this raises
a question I have never investigated. What kind of Ramsey theory gets
incomparably easier - logically and in the sense of upper bounds - if
we assume that the data is random? Would this apply to this kind of
unusually strong Ramsey theory associated with PA? And what are the
prospects are really proving suitable randomness of the lower
quesidue?
On another note, it does seem somewhat intricate to give a proper
formulation of Theorem 2 (of course Theorem 1 by specialization) with
"k factors" instead of "at most k factors". This seems well worth
doing, assuming this lower quesidue approach has legs. Also even
keeping "at most k factors" but raising the (1+logk)(2+logk)/2 is very
interesting, calibrating the strengths.
So this seems to be improving nicely at the PA level, looking less
like a slavish translation of combinatorics. People will probably be
interested in running statistical tests on upper and lower quesidues.
I wonder how close the number theory literature is to such notions?
Of course, it very much remains to be seen just what I come up with
that uses large cardinals.
************************************************************************
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 785th 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 #1-699 can be found at
http://u.osu.edu/friedman.8/foundational-adventures/fom-email-list/
700: Large Cardinals and Continuations/14 8/1/16 11:01AM
701: Extending Functions/1 8/10/16 10:02AM
702: Large Cardinals and Continuations/15 8/22/16 9:22PM
703: Large Cardinals and Continuations/16 8/26/16 12:03AM
704: Large Cardinals and Continuations/17 8/31/16 12:55AM
705: Large Cardinals and Continuations/18 8/31/16 11:47PM
706: Second Incompleteness/1 7/5/16 2:03AM
707: Second Incompleteness/2 9/8/16 3:37PM
708: Second Incompleteness/3 9/11/16 10:33PM
709: Large Cardinals and Continuations/19 9/13/16 4:17AM
710: Large Cardinals and Continuations/20 9/14/16 1:27AM
700: Large Cardinals and Continuations/14 8/1/16 11:01AM
701: Extending Functions/1 8/10/16 10:02AM
702: Large Cardinals and Continuations/15 8/22/16 9:22PM
703: Large Cardinals and Continuations/16 8/26/16 12:03AM
704: Large Cardinals and Continuations/17 8/31/16 12:55AM
705: Large Cardinals and Continuations/18 8/31/16 11:47PM
706: Second Incompleteness/1 7/5/16 2:03AM
707: Second Incompleteness/2 9/8/16 3:37PM
708: Second Incompleteness/3 9/11/16 10:33PM
709: Large Cardinals and Continuations/19 9/13/16 4:17AM
710: Large Cardinals and Continuations/20 9/14/16 1:27AM
711: Large Cardinals and Continuations/21 9/18/16 10:42AM
712: PA Incompleteness/1 9/23/16 1:20AM
713: Foundations of Geometry/1 9/24/16 2:09PM
714: Foundations of Geometry/2 9/25/16 10:26PM
715: Foundations of Geometry/3 9/27/16 1:08AM
716: Foundations of Geometry/4 9/27/16 10:25PM
717: Foundations of Geometry/5 9/30/16 12:16AM
718: Foundations of Geometry/6 101/16 12:19PM
719: Large Cardinals and Emulations/22
720: Foundations of Geometry/7 10/2/16 1:59PM
721: Large Cardinals and Emulations//23 10/4/16 2:35AM
722: Large Cardinals and Emulations/24 10/616 1:59AM
723: Philosophical Geometry/8 10/816 1:47AM
724: Philosophical Geometry/9 10/10/16 9:36AM
725: Philosophical Geometry/10 10/14/16 10:16PM
726: Philosophical Geometry/11 Oct 17 16:04:26 EDT 2016
727: Large Cardinals and Emulations/25 10/20/16 1:37PM
728: Philosophical Geometry/12 10/24/16 3:35PM
729: Consistency of Mathematics/1 10/25/16 1:25PM
730: Consistency of Mathematics/2 11/17/16 9:50PM
731: Large Cardinals and Emulations/26 11/21/16 5:40PM
732: Large Cardinals and Emulations/27 11/28/16 1:31AM
733: Large Cardinals and Emulations/28 12/6/16 1AM
734: Large Cardinals and Emulations/29 12/8/16 2:53PM
735: Philosophical Geometry/13 12/19/16 4:24PM
736: Philosophical Geometry/14 12/20/16 12:43PM
737: Philosophical Geometry/15 12/22/16 3:24PM
738: Philosophical Geometry/16 12/27/16 6:54PM
739: Philosophical Geometry/17 1/2/17 11:50PM
740: Philosophy of Incompleteness/2 1/7/16 8:33AM
741: Philosophy of Incompleteness/3 1/7/16 1:18PM
742: Philosophy of Incompleteness/4 1/8/16 3:45AM
743: Philosophy of Incompleteness/5 1/9/16 2:32PM
744: Philosophy of Incompleteness/6 1/10/16 1/10/16 12:15AM
745: Philosophy of Incompleteness/7 1/11/16 12:40AM
746: Philosophy of Incompleteness/8 1/12/17 3:54PM
747: PA Incompleteness/2 2/3/17 12:07PM
748: Large Cardinals and Emulations/30 2/15/17 2:19AM
749: Large Cardinals and Emulations/31 2/15/17 2:19AM
750: Large Cardinals and Emulations/32 2/15/17 2:20AM
751: Large Cardinals and Emulations/33 2/17/17 12:52AM
752: Emulation Theory for Pure Math/1 3/14/17 12:57AM
753: Emulation Theory for Math Logic 3/10/17 2:17AM
754: Large Cardinals and Emulations/34 3/12/17 12:34AM
755: Large Cardinals and Emulations/35 3/12/17 12:33AM
756: Large Cardinals and Emulations/36 3/24/17 8:03AM
757: Large Cardinals and Emulations/37 3/27/17 2:39AM
758: Large Cardinals and Emulations/38 4/10/17 1:11AM
759: Large Cardinals and Emulations/39 4/10/17 1:11AM
760: Large Cardinals and Emulations/40 4/13/17 11:53PM
761: Large Cardinals and Emulations/41 4/15/17 4:54PM
762: Baby Emulation Theory/Expositional 4/17/17 1:23AM
763: Large Cardinals and Emulations/42 5/817 2:18AM
764: Large Cardinals and Emulations/43 5/11/17 12:26AM
765: Large Cardinals and Emulations/44 5/14/17 6:03PM
766: Large Cardinals and Emulations/45 7/2/17 1:22PM
767: Impossible Counting 1 9/2/17 8:28AM
768: Theory Completions 9/4/17 9:13PM
769: Complexity of Integers 1 9/7/17 12:30AM
770: Algorithmic Unsolvability 1 10/13/17 1:55PM
771: Algorithmic Unsolvability 2 10/18/17 10/15/17 10:14PM
772: Algorithmic Unsolvability 3 Oct 19 02:41:32 EDT 2017
773: Goedel's Second: Proofs/1 Dec 18 20:31:25 EST 2017
774: Goedel's Second: Proofs/2 Dec 18 20:36:04 EST 2017
775: Goedel's Second: Proofs/3 Dec 19 00:48:45 EST 2017
776: Logically Natural Examples 1 12/21 01:00:40 EST 2017
777: Goedel's Second: Proofs/4 12/28/17 8:02PM
778: Goedel's Second: Proofs/5 12/30/17 2:40AM
779: End of Year Claims 12/31/17 8:03PM
780: One Dimensional Incompleteness/1 1/4/18 1:14AM
781: One Dimensional Incompleteness/2 1/6/18 11:25PM
782: Revolutionary Possibilities/1 1/12/18 11:26AM
783: Revolutionary Possibilities/2 1/20/18 9:43PM
784: Revolutionary Possibilities/3 1/21/18 2:59PM
Harvey Friedman
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