[FOM] 381:Trigonometric Block Theorems

[Harvey Friedman]

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
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 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