[FOM] 251:Embedding Axioms

[Harvey Friedman]

EMBEDDING AXIOMS
by
Harvey M. Friedman
July 31, 2005

We give axioms in the style of Vopenka's Principle which are much stronger.
They follow from I2 and imply the existence of lots of ranks that have
nontrivial elementary embeddings into themselves.

1. Vopenka's Principle.

This is usually stated as an axiom over NBG + AxC as follows:

VP. Let X be a proper class of structures of the same relational type. There
exist distinct A,B in X such that A is elementarily embeddable into B.

This uses arbitrary (set) relational types. Obviously finite relational
types are more common.

VP1. Let X be a proper class of structures of the same finite relational
type. There exist distinct A,B in X such that A is elementarily embeddable
into B. 

Also, why use elementarity? It is more "elementary" to avoid "elementarity".

VP2. Let X be a proper class of structures of the same finite relational
type. There exist distinct A,B in X such that A is embeddable into B.

It is more algebraic to use a single binary operation:

VP3. Let X be a proper class of binary operations. There exist distinct A,B
in X such that A is embeddable into B.

It is arguably more "elementary" to instead use binary relations. Here a
binary relation is merely a set of ordered pairs.

VP4. Let X be a proper class of binary relations. There exist distinct A,B
in X such that A is embeddable into B.

Why not use irreflexive symmetric binary relations - like graphs without
loops and without isolated points?

VP5. Let X be a proper class of irreflexive symmetric binary relations.
There exist distinct A,B in X such that A is embeddable into B.

Why not use positive embeddings, as in graph embedding? I.e., let R,S be
binary relations. A positive embedding from R into S is a one-one function
f:fld(R) into fld(S) such that R(x,y) implies S(fx,fy). Here we use implies
rather than iff. 

VP6. Let X be a proper class of irreflexive symmetric binary relations.
There exist distinct A,B in X such that A is positively embeddable into B.

THEOREM 1.1. NBG + AxC proves that VP,VP1 - VP6 are all equivalent.

THEOREM 1.2. The following are equiconsistent.
i. NBG + AxC + any of VP,VP1 - VP6.
ii. NBG + AxC + "for all proper X containedin On, there exists alpha < beta
from X such that V(alpha) is nontrivially elementarily embeddable into
V(beta)".
iii. ZFC + any of VP,VP1 - VP6 as a scheme in the language of set theory.
iv. ZFC + "for all proper X containedin On given by an explicit definition
with parameters, there exists alpha < beta from X such that V(alpha) is
nontrivially elementarily embeddable into V(beta)".
Furthermore, i implies ii implies iii implies iv, and iii,iv are equivalent.

2. New Principle over NBG + AxC.

$1. Every proper class of relations of the same finite arity has a subset
(countable subset) (subclass) whose union is nontrivially self embeddable.

$1 clearly has 3 forms.

$2. Every proper class of irreflexive symmetric binary relations has a
subset (countable subset) (subclass) whose union is nontrivially positively
self embeddable. 

$2 clearly has 3 forms.

We take the official form of $ to be

$. Every proper class of relations of the same finite arity has a subset
whose union is nontrivially self embeddable.

THEROEM 2.1. NBG + AxC proves that all of the above six forms of $ are
equivalent.

THEOREM 2.2. The following are equiconsistent.
i. NBG + AxC + any of the six forms of $.
ii. NBG + AxC + "for all closed unbounded X containedin On, there exists
alpha in X such that V(alpha) is nontrivially elementarily self embeddable".
iii. ZFC + any of the six forms of $ as a scheme in the language of set
theory. 
iv. ZFC + "for all closed unbounded X containedin On given by an explicit
definition with parameters, there exists alpha in X such that V(alpha) is
nontrivially elementarily self embeddable".
Furthermore, i implies ii implies iii implies iv, and iii,iv are equivalent.

3. Localized forms.

The following definition is standard. We say that kappa is a Vopenka
cardinal if and only if VP holds in V(kappa + 1).

THEOREM 3.1. ZFC proves that the following are equivalent.
i. kappa is Vopenka.
ii. Let X be a cardinality kappa set of binary operations, each of which is
of cardinality < kappa. There exist distinct A,B in X such that A is
embeddable into B.
iii. Let X be a cardinality kappa set of irreflexive symmetric binary
relations, each of which is of cardinality < kappa. There exist distinct A,B
in X such that A is positively embeddable into B.

We say that a cardinal kappa has property $) if and only if the following
holds. 

Let X be a cardinality kappa set of relations of the same finite arity, each
of which is of cardinality < kappa. There exists a subset of X whose union
is nontrivially self embeddable.

I.e., kappa is $) if and only if $ holds in V(kappa + 1).

THEOREM 3.2. ZFC proves that the following are equivalent.
i. kappa is $.
ii. Let X be a cardinality kappa set of irreflexive symmetric binary
relations, each of which is of cardinality < kappa. There exists a subset of
X whose union is nontrivially positively self embeddable.

It is well known that

i. If kappa is almost huge then kappa is Vopenka. Also {alpha < kappa: alpha
is Vopenka} is stationary.
ii. If kappa is Vopenka, then {alpha < kappa: alpha is extendible} is
stationary in kappa.

We can show that 

iii. if kappa is I2 then {alpha < kappa: alpha is $)} is stationary in
kappa. 
iv. If kappa is $ then {alpha < kappa: V(alpha) is nontrivially elementarily
self embeddable} is stationary in kappa.

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

I use www.math.ohio-state.edu/~friedman/ for downloadable manuscripts.
This is the 250th 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 #1-149 can be found at
http://www.cs.nyu.edu/pipermail/fom/2005-June/008999.html in the FOM
archives, 6/15/05, 9:18PM.

250. Extreme Cardinals/Pi01  7/31/05  8:34PM

Harvey Friedman