TITLE: Is it possible to formalize the idea that large cardinal axioms don't limit us, while their negations do?
QUESTION [5 upvotes]: This question has been substantially rewritten. Hopefully, its clearer now.
Suppose our goal is to describe every material set that can exist by extending ZFC. Of course, this is an impossible goal, since a consistent extension of ZFC cannot prove the existence of a model of its own axioms. However we can certainly make progress towards describing everything that can exist, especially by adjoining large cardinal axioms.
Now suppose that to ZFC we add "There exists an inaccessible uncountable cardinal." Intuitively, we haven't really limited ourselves in any way, with respect to the goal of describing every set that can exist. On other hand, suppose that to ZFC we add "There does not exist an inaccessible uncountable cardinal." Well, if our goal is to describe every material set that can exist, then I think adding this axiom is a very bad move.
In general, adding a large cardinal axiom $\phi$ to ZFC okay (so long as $\phi$ is consistent, etc.) while adding $\neg \phi$ is a bad move. Is it possible to formalize this idea, that large cardinal axioms don't limit us, while their negations do?

REPLY [3 votes]: This topic is both mathematically and philosophically rich. What precisely do we mean when we say that a set-theoretic principle is "restrictive"? Can we give a formal account of this notion that aligns with our prereflective intuitions about restrictiveness? This is how I understand your question.
Penelope Maddy attempted to do exactly that in her book Naturalism in Mathematics, following up on her article, V=L and Maximize, in which she made a very specific technical proposal of what it is that makes a theory formally `restrictive'. The aim is to reify the informal concept of restrictiveness that is present in your question, and which was present in Maddy's earlier accounts in the links that Andres cites, so that we may treat restrictivenes as a mathematically formal notion.
Was the proposal successful? Well, you'll have to judge for yourself. The proposal has been criticized from a number of angles, including B. Loewe, A first glance at non-restrictiveness, and B. Loewe, A second glance at non-restrictiveness, where he proves among other things that ZFC itself is formally restrictive on Maddy's account. My article A multiverse perspective on the axiom of constructibility finds further serious problematic issues. In the end, it seems to me that one will need to revise the proposal, if a satisfactory formal account of restrictiveness is desired.
Meanwhile, in my paper I also address head-on the purely intuitive argument Maddy and others make against the axiom of constructibility — what I call the V≠L via maximize position — which rejects the axiom of constructibility V=L on the basis that it is restrictive. After first explaining various senses in which $V=L$ is compatible with set-theoretic strength (for example, every countable model of ZFC has an end-extension to a model of $V=L$), I argue that the $V\neq L$ via maximize position implicitly takes a stand in the pluralist debate in the philosophy of set theory by presuming an absolute background concept of ordinal. In particular, the $V\neq L$ via maximize argument appears to lose its force, in contrast, on an upwardly extensible concept of set, in light of the various facts showing that models of set theory generally have extensions to models of V=L inside larger set-theoretic universes. That is, with a robust multiverse concept of set, we no longer see $V=L$ as limiting, since larger universes can have large cardinals again, and still larger universes recover $V=L$ again. This perspective also avoids Steel's objections on the instrumentalist dodge, which are similarly based on an absolute background concept of ordinal.
You can read the paper, but here is the concluding paragraph:

Ultimately, the multiverse vision entails an upwardly extensible concept of set, where any current set-theoretic universe may be extended to a much larger, taller universe. The
current universe becomes a countable model inside a larger universe, which has still larger extensions, some with large cardinals, some without, some with the continuum hypothesis, some without, some with $V=L$ and some without, in a series of further extensions continuing longer than we can imagine. Models that seem to have $0^\sharp$ are extended to larger models where that version of $0^\sharp$ no longer works as $0^\sharp$, in light of the new ordinals. Any given set-theoretic situation is seen as fundamentally compatible with $V=L$, if one is willing to make the move to a better, taller universe. Every set, every universe of sets, becomes both countable and constructible, if we wait long enough. Thus, the constructible universe $L$ becomes a rewarder of the patient, revealing hidden constructibility structure for any given mathematical object or universe, if one should only extend the ordinals far enough beyond one's current set-theoretic universe. This perspective turns the $V\neq L$ via maximize argument on its head, for by maximizing the ordinals, we seem able to recover $V=L$ as often as we like, extending our current universe to larger and taller universes in diverse ways, attaining $V=L$ and destroying it in an on-again, off-again pattern, upward densely in the set-theoretic multiverse, as the ordinals build eternally upward, eventually exceeding any particular conception of them.

For your question, the point is that negated large cardinal axioms similarly needn't be viewed as limiting, if we think there is a much larger universe that has large cardinals, even if that universe is countable inside a model of $V=L$, in an endless tower of universes.