[FOM] 503: Proper Classes of Relations

Sat Oct 13 12:17:36 EDT 2012

THIS RESEARCH WAS PARTIALLY SUPPORTED BY THE JOHN TEMPLETON FOUNDATION

*****************************************

THIS POSTING IS ENTIRELY SELF CONTAINED

*****************************************

PROPER CLASSES OF GRAPHS
by
Harvey M. Friedman
October 13, 2012

Abstract. The Graph Embedding Principle is an elemental version of
Vopenka's Principle for graphs. The Graph Self Embedding Principle is
a variant of the Graph Embedding Principle. GEP is equivalent to
Vopenka's Principle. GSEP implies I1 and is weaker than I2.

1. GRAPH EMBEDDING PRINCIPLE.

A graph is a pair (V,E), where V is a set and E contained in V^2 is an
irreflexive symmetric relation.

We say that f is an embedding from G = (V,E) to G' = (V',E') if and only if

i. f:V into V' is one-one.
ii. If E(x,y) then E'(f(x),f(y)).

We say that f is a strong embedding from G to G' if and only if

i. f:V into V' is one-one.
ii. E(x,y) if and only if E'(f(x),f(y)).

We say that f is nontrivial if and only if f is not an identity
function. We say that f is proper if and only if the range of f is a
proper subset of its domain.

GRAPH EMBEDDING PRINCIPLE. GEP. TWELVE FORMS. In every proper class of
graphs, some element is

(1) embeddable, (2) strongly embeddable, (3) nontrivially embeddable,
(4) properly embeddable 5) nontrivially strongly embeddable 6)
properly strongly embeddable

into

(a) an element, (b) a different element.

THEOREM (NBG). The following are equivalent.
i. any form of GEP with the exception of 1a, 2a.
ii. Vopenka's Principle.

2. GRAPH SELF EMBEDDING PRINCIPLE.

The union of graphs is formed by using the union of the vertices and
the union of the edges.

GRAPH SELF EMBEDDING PRINCIPLE. GSEP. SIX FORMS. In every proper class
of graphs, the union of some

(1) subset, (2) infinite subset, (3) countably infinite subset

is

(a) nontrivially, (b) properly

self embeddable.

THEOREM 2.1.(NBG). All six forms of GSEP imply all forms of GEP, and
I3. I2 implies that all six forms of GSEP hold in V(kappa+1), where
kappa is the critical point of any j given by I2.

We remark that these results hold even if we formulate NBG without any
axiom of choice.

*****************************************

I use http://www.math.ohio-state.edu/~friedman/ for downloadable
manuscripts. This is the 503rd 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  4/5/12  4:17PM
495: Invariant Finite Choice  4/5/12  4:18PM
496: Invariant Finite Choice/restatement  4/8/12  2:18AM
497: Invariant Maximality Restated  5/2/12 2:49AM
498: Embedded Maximal Cliques 1  9/18/12  12:43AM
499. Embedded Maximal Cliques 2  9/19/12  2:50AM
500: Embedded Maximal Cliques 3  9/20/12  10:15PM
501: Embedded Maximal Cliques 4  9/23/12  2:16AM
502: Embedded Maximal Cliques 5  9/26/12  1:21AM

Harvey Friedman