I wonder if it's possible to formulate an "ultimate" maximality principle (UMP) and prove its consistency. I envision UMP to express the idea that no matter how we enlarge the universe of set theory V (by any means e.g. set forcing, class forcing, infinite model theory), we would gain n o t h i n g. Let W be the ultimate enlargement of V. Then UMP would say that a statement is true in W iff it's true in V. So any statement that is true in W is already true in V.


Questions:
1) Are there available reference in literature concerning UMP?
2) If not, what is the prospect of UMP in foundational research?



Ultimate Maximality Principle
There are several maximality principles that already have
some of this flavor, with a growing literature surrounding them.


For example, the maximality principle MP as introduced in
my paper [A simple maximality
principle](http://jdh.hamkins.org/maximalityprinciple/),
and also in a paper of Stavi and Vanaanan, is the scheme
asserting that any statement $\varphi$ that is forceable by
set forcing in such a way that it remains true in all
further set forcing extensions, is already true in $V$.
This axiom MP is actually equiconsistent with ZFC. Stronger
versions of the axiom allow countable parameters (the axiom
is inconsistent with uncountable parameters, since they can
become countable by forcing). The strongest version of the
axiom is the Necessary Maximality Principle NMP, which
asserts that $\text{MP}(\mathbb{R})$ holds in all set forcing
extensions, and this has determinacy consequences, but has
strength below $\text{AD}\_{\mathbb{R}}+\Theta$ is regular.
The natural analogue of MP for class forcing is either
inconsistent or not expressible in first order set theory.


Another tack on the issue is the [Inner Model
Hypothesis](http://arxiv.org/abs/0711.0680) of Sy Friedman,
which aspires more in the universal direction of your
question. Namely, the IMH asserts that if there is any
outer model of the universe having an inner model
satisfying a certain assertion, then there is already such
an inner model with that feature. This axiom has the flavor
of what you have wanted; it has a strong consistency
strength, but it itself is inconsistent with the actual
existence of large cardinals, as opposed to their existence
in inner models. The penalty for the greater universality
of the IMH is that it is not expressible in first order set
theory as an axiom about $V$. One can, however, understand
it as an external assertion about a countable model,
treating that countable model as a kind of universe
substitute.


Both the MP and the IMH are naturally expressible in modal
terms by the S5 axiom
$\Diamond\square\varphi\to\square\varphi$, which expresses
the idea that anything that could become necessarily true
is already necessarily true. Benedikt Loewe and I explored
the nature of the forcing modality in our paper [The modal
logic of
forcing](http://jdh.hamkins.org/themodallogicofforcing/).


Your proposed Ultimate Maximality Principle would seem to
need a more detailed fleshing out: in the axiom you refer
to an "ultimate" enlargement $W$ of $V$, but what is this
$W$? After all, for any enlargement $W$ of $V$ we may
continue to form the forcing extensions of $W$, so strictly
speaking, there is no largest one. For example, $W$ itself
would have forcing extensions, some with CH and others with
$\neg\text{CH}$. Similarly, any set in $W$ can be made
countable by forcing, and so if you are entertaining the
idea of a single largest one, then every set in $V$ would
have to be countable in it. So the idea that one can
achieve literal maximality as you describe becomes
problematic, and this is the reason why the MP and the IMH
make use of the S5 style maximality, which asserts that
anything that could become true forever afterwards is
already true, an assertion that takes the place of an
actual maximal extension.


Meanwhile, there is current work to investigate the extent
to which we may have maximality-type principles for class
forcing and for arbitrary extensions. For example, it
appears that one may get it for extensions with the
approximation and cover properties without much
modification from the original work.


