Ultimate Maximality Principle

Question

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?

Answer 1

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, 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 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.

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.