1
RESTRICTIONS AND EXTENSIONS
by
Harvey M. Friedman
Ohio State University
Princeton University
[email protected]
http://www.math.ohiostate.edu/~friedman/
February 17, 2003
Abstract. We consider a number of statements involving
restrictions and extensions of algebras, and derive
connections with large cardinal axioms.
1. Introduction.
By an algebra M we mean a nonempty set dom(M) together with
a finite list of functions on dom(M) of various finite
arities
≥
0. The signature of M is the list of arities of
the functions of M.
Let M,N be algebras. We say that M is a restriction of N if
and only if M,N have the same signature, dom(M) dom(N),
and the functions of M are restrictions of the respective
functions of N. We say that N is an extension of M if and
only if M is a restriction of N.
We use
k
,
l
for cardinals throughout the paper.
We write R(
k
,fg) if and only if every algebra of cardinality
k
has a proper restriction with the same finitely generated
restrictions up to isomorphism.
We write R(
k
,
l
) if and only if every algebra of cardinality
k
has a proper restriction with the same restrictions of
cardinality
l
up to isomorphism.
We write E(
k
,fg) if and only if every algebra of cardinality
k
has a proper extension with the same finitely generated
restrictions up to isomorphism.
We write E(
k
,
l
) if and only if every algebra of cardinality
k
has a proper extension with the same restrictions of
cardinality
l
up to isomorphism.
In this paper, we will restrict attention to the cases
l
=
w
. All theorems are proved in ZFC.
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R(
k
,fg), R(
k
,
w
) are easily treated as follows.
THEOREM 1.1. The following are equivalent.
i) R(
k
,fg);
ii) R(
k
,
w
);
iii)
k
> 2
w
.
Our main results concern sufficiently large
k
.
THEOREM 1.2. The following are equivalent.
ii) for all sufficiently large
k
, E(
k
,fg);
ii) for all sufficiently large
k
, E(
k
,
w
);
iii) there exists a measurable cardinal.
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 Math, Model theory, Continuum hypothesis, ZFC, large cardinals

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