[FOM] The width of V
Roger Bishop Jones
rbj at rbjones.com
Tue Jul 8 06:31:44 EDT 2003
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
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