Editing Large cardinal (section)

==Motivations and epistemic status==
Large cardinals are understood in the context of the [[von Neumann universe]] V, which is built up by [[transfinite induction|transfinitely iterating]] the [[powerset]] operation, which collects together all [[subset]]s of a given set. Typically, [[Model theory|models]] in which large cardinal axioms ''fail'' can be seen in some natural way as submodels of those in which the axioms hold. For example, if there is an [[inaccessible cardinal]], then "cutting the universe off" at the height of the first such cardinal yields a [[universe (mathematics)|universe]] in which there is no inaccessible cardinal. Or if there is a [[measurable cardinal]], then iterating the ''definable'' powerset operation rather than the full one yields [[Gödel's constructible universe]], L, which does not satisfy the statement "there is a measurable cardinal" (even though it contains the measurable cardinal as an ordinal).

Thus, from a certain point of view held by many set theorists (especially those inspired by the tradition of the [[Cabal (set theory)|Cabal]]), large cardinal axioms "say" that we are considering all the sets we're "supposed" to be considering, whereas their negations are "restrictive" and say that we're considering only some of those sets. Moreover the consequences of large cardinal axioms seem to fall into natural patterns (see Maddy, "Believing the Axioms, II"). For these reasons, such set theorists tend to consider large cardinal axioms to have a preferred status among extensions of ZFC, one not shared by axioms of less clear motivation (such as [[Martin's axiom]]) or others that they consider intuitively unlikely (such as [[V=L|V&nbsp;=&nbsp;L]]). The hardcore [[Philosophy of mathematics#Mathematical realism|realists]] in this group would state, more simply, that large cardinal axioms are ''true''.

This point of view is by no means universal among set theorists. Some [[Philosophy of mathematics#Formalism|formalists]] would assert that standard set theory is by definition the study of the consequences of ZFC, and while they might not be opposed in principle to studying the consequences of other systems, they see no reason to single out large cardinals as preferred. There are also realists who deny that [[ontological maximalism]] is a proper motivation, and even believe that large cardinal axioms are false. And finally, there are some who deny that the negations of large cardinal axioms ''are'' restrictive, pointing out that (for example) there can be a [[transitive set]] model in L that believes there exists a measurable cardinal, even though L itself does not satisfy that proposition.