[FOM] The width of V

[Roger Bishop Jones]

On Tuesday 08 July 2003  3:55 am, Kanovei wrote:
> From: Roger Bishop Jones <rbj at rbjones.com>
>
> >Can anyone point me to work on axioms which
>
> are intended to make V as fat as possible?
>
> Martin's axiom MA makes continuum rather fat because
> it implies that holes of certain kind are fulfilled.

Can one be sure that an axiom is fattening?

An axiom of the form:

	forall x such that P(x) there exists y such that Q(x,y)

looks superficially like it can only add sets,
but since it is equivalent to:

	not P(x) \/ exists y such that Q(x,y)

and if P(x) has the form:

	there exists z such that R(x,z)

the axiom might just as well deny existence as assert it.

To resolve this kind of question one would need
a definition of "fattening", do you know whether anyone
has ever attempted such a thing?

Suppose I moot the definition:

An axiom A is fattening iff:

 forall alpha, M where M |= Z and rank(M)= alpha
 there exists M' of rank alpha containing M
	such that M' |= Z+A

(This is intended to say that A only eliminates thin
models.)

Does this sound like a reasonable definition of
"fattening" and do you think Martin's axiom could
be shown to be fattening in this sense?

Roger Jones