[FOM] Re: The width of V

[E. Todd Eisworth]

I'd like to add another question to Roger Bishop's line of inquiry.  
It seems to me that a lot of attention on "fattening the universe" has
focused on the neighborhood of $\omega_1$ and the continuum.  (I guess this
is to be expected as this is the neighborhood where most mathematicians
live.)

A major theme seems to be "filling in holes in the universe" (Woodin does
this by P_max, Foreman-Magidor-Shelah by iterating semi-proper forcing).
Does this "fattening up" of low levels of the universe necessitate "weight
loss" at higher levels of the cumulative hierarchy?

For example, there are forcing axioms in the spirit of MA(\omega_1) and the
Proper Forcing Axiom that are consistent with the Continuum Hypothesis.
There are also generalized versions of Martin's Axiom that apply to certain
$\aleph_1$--closed $\aleph_2$--chain condition notions of forcing. Both
sorts of axioms attempt to "fill in holes", and each (individually) is
consistent with ZFC + CH.  However, we cannot get such axioms to hold
simultaneously in a model of CH:  strong enough axioms of the first sort
preclude the existence of Kurepa trees, while a relatively weak form of the
second sort implies that they exist.

[References: Chapter VII of Shelah's Proper and Improper forcing for the
first sort of axiom, "Some applications of a generalized Martin's Axiom" by
Frank Tall for the second sort of axiom.]


Does this sort of phenomenon arise in the context of Woodin's work or other
attempts to fatten up the universe at low levels?



Todd