[FOM] Re: Definition of "large cardinal axiom"?

Ali Enayat enayat at american.edu
Wed Apr 14 15:07:02 EDT 2004

This is a reply to Roger Jones, who has asked for a definition of "large
cardinal axiom".

I will give a operative definition in the ZF context:

(1) There are a family of statementts, ranging in strength from "there
exists an inaccessible cardinal", to "there exists an elementary embedding
of V into a subclass M with such and such closure properties" that are
known as "large cardinal axioms" [see, e.g. Kanamori's THE HIGHER INFINITE,
or Jech's SET THEORY].

(2) Some mathematical statements might be *equivalent* in ZFC to a large
cardinal axiom as in (1), and be thus referred to as a large cardinal
hypothesis.  For example, the statement "there is a sigma-additive measure
mu(X) on the power-set of some infinite set X such that mu({x})=0 for all x
in X, and mu(X)=1" is equivalent to the statement "there is a measurable
cardinal" [thanks to Ulam's work in 1930].

(3) Some mathematical statements might *imply* a large cardinal axiom as in
(1), or they might imply the truth of a large cardinal axiom in some inner
model of set theory (such as Godel's constriuctible universe L). Often such
statements are also referred to as a large cardinal axiom. For example, the
statement "all subsets of reals have the property of Baire" is known to
imply that "there is an inaccessible cardinal in L" (thanks to a theorem of
Shelah in 1980).

(4) Finally, some statements S are referred to as large cardinal axioms if
Con(ZFC + a large cardinal axiom in the sense of (1)) follows from
Con(ZFC+S). This is the most liberal definition of a "large cardinal
axiom". [Here Con(T) is "T is consistent", or equivalently "T has a
For example, the statement S = "the theory ZFC+ O-sharp exists is
consistent" implies
Con(ZFC + there is a weakly compact cardinal).

Ali Enayat
Department of Mathematics and Statistics
American University
4400 Massachusetts Ave, NW
 Washington, DC 20016-8050
(202) 885-3168

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