# [FOM] Definition of "large cardinal axiom"?

Thomas Forster T.Forster at dpmms.cam.ac.uk
Wed Apr 14 05:31:34 EDT 2004



Roger,
Perhaps i should, but i can't forbear to quote myself....

Set theories with $V \in V$ retain some ancestral features lost in the
more specialized theories of well-founded sets. This becomes apparent when
we consider the possibility of standard versions of strong axioms.  A
strong axiom is one that says that $V$ is closed under some operation, or
that something highly improbable happens.  Either way it boils down to
saying that the universe is large, and since ZF nails it down at one end,
the other is a long way away. Thus in well-founded theories like ZF -
whatever we started off trying to say - the result is always liable to be
(equivalent to) a {\sl large cardinal axiom}. The fact that interesting
allegations in ZF tend to be related to large cardinal axioms is thus
revealed to be merely an artefact of the axiom of foundation, which
determines that large cardinals are, so to speak, the local currency in
which information is traded and strength is denominated. The situation in
set theories with $V \in V$ is not so straightforward.

On Tue, 13 Apr 2004, Roger Bishop Jones wrote:

> Can anyone tell me whether there is a generally accepted
> definition of what is a large cardinal axiom?
>
> None of the texts which I have to hand seems to have a definition
> of "large cardinal axiom" in general, though they have definitions
> of various particular kinds of large cardinal, i.e. they
> have examples of large cardinal axioms.
>
> I do have a paper by Patrick Dehornoy (Progres recents sur
> l'hypothese du continu) which has a definition in it (Def. 6.1)
> but it looks as if this definition may have been devised
> for the purposes of the work in hand, rather than being
> a generally accepted definition.
>
> Roger Jones
>
> - rbj01 at rbjones.com
>    plain text email please (non-executable attachments are OK)
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