[FOM] 448: Naturalness/PA Independence
Harvey Friedman
friedman at math.ohio-state.edu
Thu Dec 2 03:30:29 EST 2010
THIS RESEARCH WAS PARTIALLY SUPPORTED BY THE JOHN TEMPLETON FOUNDATION
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I recently discussed the following three theorems with a famous
combinatorialist in terms of mathematical naturalness. It would be
interesting to see what readers have to say about this.
THEOREM 1. Let n >> k,r,p >= 1. Every r coloring of the unordered k-
tuples from {1,...,n} has an at least p element monochromatic set with
at least as many elements as its min.
THEOREM 2. Let n >> k >= 1. For every x_1,...,x_n from {1,...,k},
there exists 1 <= i < j <= n/2 such that x_i,...,x_2i is a subsequence
of x_j,...,x_2j.
THEOREM 3. Let n >> k >= 1. For every f:{1,...,n}^k into {1,...,n}^k
obeying max(f(x)) <= max(x), there exist distinct x_1,...,x_k+1 such
that f(x_1,...,x_k) <= f(x_2,...,x_k+1) coordinatewise.
Theorems 1,3 are equivalent to 1-Con(PA), and Theorem 2 is equivalent
to 1-Con(PA_2). PA_2 is 2 quantifier induction.
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I use http://www.math.ohio-state.edu/~friedman/ for downloadable
manuscripts. This is the 448th 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 #1-349 can be found athttp://www.cs.nyu.edu/pipermail/fom/2009-August/014004.html
in the FOM archives.
350: one dimensional set series 7/23/09 12:11AM
351: Mapping Theorems/Mahlo/Subtle 8/6/09 10:59PM
352: Mapping Theorems/simpler 8/7/09 10:06PM
353: Function Generation 1 8/9/09 12:09PM
354: Mahlo Cardinals in HIGH SCHOOL 1 8/9/09 6:37PM
355: Mahlo Cardinals in HIGH SCHOOL 2 8/10/09 6:18PM
356: Simplified HIGH SCHOOL and Mapping Theorem 8/14/09 9:31AM
357: HIGH SCHOOL Games/Update 8/20/09 10:42AM
358: clearer statements of HIGH SCHOOL Games 8/23/09 2:42AM
359: finite two person HIGH SCHOOL games 8/24/09 1:28PM
360: Finite Linear/Limited Memory Games 8/31/09 5:43PM
361: Finite Promise Games 9/2/09 7:04AM
362: Simplest Order Invariant Game 9/7/09 11:08AM
363: Greedy Function Games/Largest Cardinals 1
364: Anticipation Function Games/Largest Cardinals/Simplified 9/7/09
11:18AM
365: Free Reductions and Large Cardinals 1 9/24/09 1:06PM
366: Free Reductions and Large Cardinals/polished 9/28/09 2:19PM
367: Upper Shift Fixed Points and Large Cardinals 10/4/09 2:44PM
368: Upper Shift Fixed Point and Large Cardinals/correction 10/6/09
8:15PM
369. Fixed Points and Large Cardinals/restatement 10/29/09 2:23PM
370: Upper Shift Fixed Points, Sequences, Games, and Large Cardinals
11/19/09 12:14PM
371: Vector Reduction and Large Cardinals 11/21/09 1:34AM
372: Maximal Lower Chains, Vector Reduction, and Large Cardinals
11/26/09 5:05AM
373: Upper Shifts, Greedy Chains, Vector Reduction, and Large
Cardinals 12/7/09 9:17AM
374: Upper Shift Greedy Chain Games 12/12/09 5:56AM
375: Upper Shift Clique Games and Large Cardinals 1graham
376: The Upper Shift Greedy Clique Theorem, and Large Cardinals
12/24/09 2:23PM
377: The Polynomial Shift Theorem 12/25/09 2:39PM
378: Upper Shift Clique Sequences and Large Cardinals 12/25/09 2:41PM
379: Greedy Sets and Huge Cardinals 1
380: More Polynomial Shift Theorems 12/28/09 7:06AM
381: Trigonometric Shift Theorem 12/29/09 11:25AM
382: Upper Shift Greedy Cliques and Large Cardinals 12/30/09 2:51AM
383: Upper Shift Greedy Clique Sequences and Large Cardinals 1
12/30/09 3:25PM
384: THe Polynomial Shift Translation Theorem/CORRECTION 12/31/09
7:51PM
385: Shifts and Extreme Greedy Clique Sequences 1/1/10 7:35PM
386: Terrifically and Extremely Long Finite Sequences 1/1/10 7:35PM
387: Better Polynomial Shift Translation/typos 1/6/10 10:41PM
388: Goedel's Second Again/definitive? 1/7/10 11:06AM
389: Finite Games, Vector Reduction, and Large Cardinals 1 2/9/10
3:32PM
390: Finite Games, Vector Reduction, and Large Cardinals 2 2/14/09
10:27PM
391: Finite Games, Vector Reduction, and Large Cardinals 3 2/21/10
5:54AM
392: Finite Games, Vector Reduction, and Large Cardinals 4 2/22/10
9:15AM
393: Finite Games, Vector Reduction, and Large Cardinals 5 2/22/10
3:50AM
394: Free Reduction Theory 1 3/2/10 7:30PM
395: Free Reduction Theory 2 3/7/10 5:41PM
396: Free Reduction Theory 3 3/7/10 11:30PM
397: Free Reduction Theory 4 3/8/10 9:05AM
398: New Free Reduction Theory 1 3/10/10 5:26AM
399: New Free Reduction Theory 2 3/12/10 9:36AM
400: New Free Reduction Theory 3 3/14/10 11:55AM
401: New Free Reduction Theory 4 3/15/10 4:12PM
402: New Free Reduction Theory 5 3/19/10 12:59PM
403: Set Equation Tower Theory 1 3/22/10 2:45PM
404: Set Equation Tower Theory 2 3/24/10 11:18PM
405: Some Countable Model Theory 1 3/24/10 11:20PM
406: Set Equation Tower Theory 3 3/25/10 6:24PM
407: Kernel Tower Theory 1 3/31/10 12:02PM
408: Kernel tower Theory 2 4/1/10 6:46PM
409: Kernel Tower Theory 3 4/5/10 4:04PM
410: Kernel Function Theory 1 4/8/10 7:39PM
411: Free Generation Theory 1 4/13/10 2:55PM
412: Local Basis Construction Theory 1 4/17/10 11:23PM
413: Local Basis Construction Theory 2 4/20/10 1:51PM
414: Integer Decomposition Theory 4/23/10 12:45PM
415: Integer Decomposition Theory 2 4/24/10 3:49PM
416: Integer Decomposition Theory 3 4/26/10 7:04PM
417: Integer Decomposition Theory 4 4/28/10 6:25PM
418: Integer Decomposition Theory 5 4/29/10 4:08PM
419: Integer Decomposition Theory 6 5/4/10 10:39PM
420: Reduction Function Theory 1 5/17/10 2:53AM
421: Reduction Function Theory 2 5/19/10 12:00PM
422: Well Behaved Reduction Functions 1 5/23/10 4:12PM
423: Well Behaved Reduction Functions 2 5/27/10 3:01PM
424: Well Behaved Reduction Functions 3 5/29/10 8:06PM
425: Well Behaved Reduction Functions 4 5/31/10 5:05PM
426: Well Behaved Reduction Functions 5 6/2/10 12:43PM
427: Finite Games and Incompleteness 1 6/10/10 4:08PM
428: Typo Correction in #427 6/11/10 12:11AM
429: Finite Games and Incompleteness 2 6/16/10 7:26PM
430: Finite Games and Incompleteness 3 6/18/10 6:14PM
431: Finite Incompleteness/Combinatorially Simplest 6/20/10 11:22PM
432: Finite Games and Incompleteness 4 6/26/10 8:39PM
433: Finite Games and Incompleteness 5 6/27/10 3:33PM
434: Digraph Kernel Structure Theory 1 7/4/10 3:17PM
435: Kernel Structure Theory 1 7/5/10 5:55PM
436: Kernel Structure Theory 2 7/9/10 5:21PM
437: Twin Prime Polynomial 7/15/10 2:01PM
438: Twin Prime Polynomial/error 9/17/10 1:22PM
439: Twin Prime Polynomial/corrected 9/19/10 2:16PM
440: Finite Phase Transitions 9/26/10 1:28PM
441: Equational Representations 9/27/10 4:59PM
442: Kernel Structure Theory Restated 10/11/10 9:01PM
443: Kernels and Large Cardinals 1 10/21/10 12:16AM
444: The Exploding Universe 1 11/1/10 1:46AMs
445: Kernels and Large Cardinals II 11/17/10 10:13PM
446: Kernels and Large Cardinals III 11/22/10 2:50PM
447: Kernels and Large Cardinals IV 11/23/10 3:51PM
Harvey Friedman
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