[FOM] 381:Trigonometric Block Theorems
Harvey Friedman
friedman at math.ohio-state.edu
Tue Dec 29 11:25:36 EST 2009
For background, see http://www.cs.nyu.edu/pipermail/fom/2009-December/014272.html
By integers, we mean Z (not N).
We use the usual notion of subsequence in analysis - i.e., we can skip
over terms, but always must move right.
A *block* is a subsequence that does not skip over terms.
A k-block is a block of length k.
Tangent here means the trigonometric tan function.
THEOREM 1. Let k >= 1. Every infinite sequence of integers contains an
infinite subsequence, where the tangents of the products of its k-
blocks lie within 1 of each other, or go to +-infinity.
We make this Theorem successively more concrete as follows.
THEOREM 2. Let k,n >= 1. Every infinite sequence of integers contains
a subsequence of length n, where the tangents of the products of its k-
blocks lie within 1 of each other, or are strictly increasing and
positive, or are strictly decreasing and negative.
THEOREM 3. Let k >= 1. Every infinite sequence of integers contains a
subsequence of length k+2, where the tangents of the products of its k-
blocks lie within 1 of each other, or are strictly increasing and
positive, or are strictly decreasing and negative.
THEOREM 4. Let k >= 1. Every sufficiently long finite sequence of
integers obeying |x[i]| <= i, i >= 1, contains a subsequence of length
k+2, where the tangents of the products of its k-blocks lie within 1
of each other, or are strictly increasing and positive, or are
strictly decreasing and negative.
THEOREM 5. Theorems 1-4 are provable in ACA' but not in ACA_0.
Theorems 1-3 are provably equivalent to "epsilon_0 is well ordered"
over RCA_0. Theorem 4 is provably equivalent to 1-Con(PA) over EFA.
The growth rate associated with Theorem 4 is epsilon_0 recursive but
grows faster than all < epsilon_0 recursive functions.
Here ACA' is ACA_0 + "for all x,n, the n-th jump of x exists".
**********************
I use http://www.math.ohio-state.edu/~friedman/ for downloadable
manuscripts. This is the 380th 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 at http://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
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371: Vector Reduction and Large Cardinals 11/21/09 1:34AM
372: Maximal Lower Chains, Vector Reduction, and Large Cardinals
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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 1
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
Harvey Friedman
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