[FOM] 495:Invariant Finite Choice

Thu Apr 5 12:50:57 EDT 2012

THIS RESEARCH WAS PARTIALLY SUPPORTED BY THE JOHN TEMPLETON FOUNDATION

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THIS POSTING IS COMPLETELY SELF CONTAINED

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There has been a breakthrough in the search for simple explicitly Pi01  
sentences independent of ZFC.

INVARIANT FINITE CHOICE
by
Harvey M. Friedman
April 5, 2012

1. Finite Choice.
2. Upper A Invariance.
3. Invariant Finite Choice.

1.FINITE CHOICE.

A binary relation R on {1,...,n}^k is regressive if and only if for  
all x,y in {1,...,n}^k, R(x,y) implies max(x) > max(y) or x = y.

A choice function for R is a function f:{1,...,n}^k into {1,...,n}^k  
such that for all x in {1,...,n}^k, R(x,f(x)).

A detached choice function for R is a choice function for R, such that  
no two distinct values are related by R.

FINITE CHOICE THEOREM. Every reflexive regressive relation on  
{1,...,n}^k has a detached choice function. Any two such have the same  
fixed points.

The finite length inductive proof is pleasantly elementary and  
educational.

2. UPPER A INVARIANCE.

Fix k,n. We say that x,y in {1,...,n}^k are order equivalent if and  
only if for all 1 <= i,j <= k, x_i < x_j iff y_i < y_j.

Let A be a subset of {1,...,n}. We say that x,y in {1,...,n}^k are  
upper A equivalent if and only if x,y are strictly increasing, and all  
coordinates not in their longest common initial subsequence lie in A.

We say that S contained in {1,...,n}^k is upper A invariant if and  
only if for all upper A equivalent x,y in {1,...,n}^k, x in S iff y in  
S.

Functions are treated as sets of ordered pairs.

3. INVARIANT FINITE CHOICE THEOREM.

We write f::U into V into W into X if and only if f maps all of U into  
V, all of V into W, and all of W into X.

INVARIANT FINITE CHOICE THEOREM. Every reflexive regressive relation  
on {1,...,n}^k has a detached choice function f::A^k into B^k into C^k  
into D^k, |A| >= log*(n)/(8k)!, with upper A invariant range.

Here log*(n) is the number of times log can be applied to n. The (8k)!  
is overkill. We'll see what we really need in due course.

THEOREM. The Invariant Finite Choice Theorem is equivalent to Con(SRP)  
over ACA'.

SRP = ZFC + {there exists lambda with the k-SRP}_k. Lambda has the k- 
SRP iff lambda is a limit ordinal, where every partition of the  
unordered k-tuples from lambda into two pieces has a stationary  
homogeneous set. ACA' = RCA_0 + for all x,n, the n-th Turing jump of x  
exists.

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I use http://www.math.ohio-state.edu/~friedman/ for downloadable
manuscripts. This is the 495th 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 #1-449 can be found
in the FOM archives at http://www.cs.nyu.edu/pipermail/fom/2010-December/015186.html

450: Maximal Sets and Large Cardinals II  12/6/10  12:48PM
451: Rational Graphs and Large Cardinals I  12/18/10  10:56PM
452: Rational Graphs and Large Cardinals II  1/9/11  1:36AM
453: Rational Graphs and Large Cardinals III  1/20/11  2:33AM
454: Three Milestones in Incompleteness  2/7/11  12:05AM
455: The Quantifier "most"  2/22/11  4:47PM
456: The Quantifiers "majority/minority"  2/23/11  9:51AM
457: Maximal Cliques and Large Cardinals  5/3/11  3:40AM
458: Sequential Constructions for Large Cardinals  5/5/11  10:37AM
459: Greedy CLique Constructions in the Integers  5/8/11  1:18PM
460: Greedy Clique Constructions Simplified  5/8/11  7:39PM
461: Reflections on Vienna Meeting  5/12/11  10:41AM
462: Improvements/Pi01 Independence  5/14/11  11:53AM
463: Pi01 independence/comprehensive  5/21/11  11:31PM
464: Order Invariant Split Theorem  5/30/11  11:43AM
465: Patterns in Order Invariant Graphs  6/4/11  5:51PM
466: RETURN TO 463/Dominators  6/13/11  12:15AM
467: Comment on Minimal Dominators  6/14/11  11:58AM
468: Maximal Cliques/Incompleteness  7/26/11  4:11PM
469: Invariant Maximality/Incompleteness  11/13/11  11:47AM
470: Invariant Maximal Square Theorem  11/17/11  6:58PM
471: Shift Invariant Maximal Squares/Incompleteness  11/23/11  11:37PM
472. Shift Invariant Maximal Squares/Incompleteness  11/29/11  9:15PM
473: Invariant Maximal Powers/Incompleteness 1  12/7/11  5:13AMs
474: Invariant Maximal Squares  01/12/12  9:46AM
475: Invariant Functions and Incompleteness  1/16/12  5:57PM
476: Maximality, CHoice, and Incompleteness  1/23/12  11:52AM
477: TYPO  1/23/12  4:36PM
478: Maximality, Choice, and Incompleteness  2/2/12  5:45AM
479: Explicitly Pi01 Incompleteness  2/12/12  9:16AM
480: Order Equivalence and Incompleteness
481: Complementation and Incompleteness  2/15/12  8:40AM
482: Maximality, Choice, and Incompleteness 2  2/19/12 7:43AM
483: Invariance in Q[0,n]^k  2/19/12  7:34AM
484: Finite Choice and Incompleteness  2/20/12  6:37AM__
485: Large Large Cardinals  2/26/12  5:55AM
486: Naturalness Issues  3/14/12  2:07PM
487: Invariant Maximality/Naturalness  3/21/12  1:43AM
488: Invariant Maximality Program  3/24/12  12:28AM
489: Invariant Maximality Programs  3/24/12  2:31PM
490: Invariant Maximality Program 2  3/24/12  3:19PM
491: Formal Simplicity  3/25/12  11:50PM
492: Invariant Maximality/conjectures  3/31/12  7:31PM
493: Invariant Maximality/conjectures 2  3/31/12  7:32PM
494: Inv Max Templates/Z+up, upper Z+ equiv

Harvey Friedman