[FOM] 490:Invariant Maximality Program 2

Sat Mar 24 15:06:14 EDT 2012

THIS RESEARCH WAS PARTIALLY SUPPORTED BY THE JOHN TEMPLETON FOUNDATION

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FEEDBACK FROM FOM SUBSCRIBERS IS REQUESTED
as to the fundamental naturalness of the
Invariant Maximality Program

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We now systematically organize the Templates for the Invariant  
Maximality Program.

We have the following eight kinds of Templates:

IM(M,{fixed,all,fcn,equiv},{fcn,equiv))

where IM abbreviates "invariant maximality", and M is a relational  
structure. To date, I have only considered M = (Q,<), and M =  
(Q[0,1],<). For the program, (Q[0,1],<) and the (Q[0,k],<) are the  
same, as the structures are isomorphic.

We also restrict each kind of Template by three numerical parameters  
k,n,m. Here k >= 1, and 0 <= n,m <= infinity, where k represents the  
dimension, and n,m represent the number of parameters used to present  
the M elementary objects. Recall that M elementary means "quantifier  
free definable over M".

Thus the full notation for the Templates are

IM(M,{fixed,all,fcn,equiv},{fcn,equiv},k,n,m).

Here are the relevant statements for the eight kinds of Templates.

IM(M,fixed,fcn,k,n,m).

Let U contained in D^2k be <= n parameter M elementary, and T:D^2k  
into D^2k be <= m parameter M elementary. U has a T invariant maximal  
square.

IM(M,fixed,equiv,k,n,m).

Let U contained in D^2k be <= n parameter M elementary, and the  
equivalence relation E on D^2k be <= m parameter M elementary. U has  
an E invariant maximal square.

IM(M,all,fcn,k,n,m).

Let T:D^2k into D^2k be <= m parameter M elementary. Every <= n  
parameter M elementary subset of D^2k has a T invariant maximal square.

IM(M,all,equiv,k,n,m).

Let the equivalence relation E on D^2k be <= m parameter M elementary.  
Every <= n parameter M elementary subset of D^2k has an E invariant  
maximal square.

IM(M,fcn,fcn,k,n,m).

Let T_1:D^2k into D^2k be <= n parameter M elementary, and T_2:D^2k  
into D^2k be <= m parameter M elementary. Every T_1 invariant subset  
of D^2k has a T_2 invariant maximal square.

IM(M,fcn,equiv,k,n,m).

Let T:D^2k into D^2k be <= n parameter elementary, and the equivalence  
relation E on D^2k be <= m parameter elementary. Every T_1 invariant  
subset of D^2k has an E invariant maximal square.

IM(M,equiv,fcn,k,n,m).

Let the equivalence relation E on D^2k be <= n parameter M elementary,  
and T:D^2k into D^2k be <= m parameter M elementary. Every E invariant  
subset of D^2k has a T invariant maximal square.

IM(M,equiv,equiv,k,n,m).

Let the equivalence relation E_1 on D^2k be <= n parameter M  
elementary, and the equivalence relation E_2 on D^2k be <= m parameter  
M elementary. Every E_1 invariant subset of D^2k has an E_2 invariant  
maximal square.

We also allow "T invariant", "T_1 invariant", and "T_2 invariant" to  
be replaced by "completely T invariant", "completely T_1 invariant",  
and "completely T_2 invariant". This is indicated by using fcn/cmplt  
as an option in the Template notation. I.e., we have

IM({fixed,all,fcn,fcn/cmplt,equiv},{fcn,fcn/cmplt,equiv},k,n,m)

for a total of fifteen different kinds of Templates.

We conjecture that all instances of all Templates, with any k,n,m, can  
be proved or refuted in SRP.

We know that some of these Templates have instances that are provable  
in SRP, but not in ZFC. These include

IM((Q,<),fixed,ANY,16,0,16).
IM((Q,<),equiv,ANY,16,0,16).

Here any 16 can be changed to any 16 <= p <= infinity.

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I use http://www.math.ohio-state.edu/~friedman/ for downloadable
manuscripts. This is the 490th 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

Harvey Friedman

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