In the mathematical field ofset theory, alarge cardinal property is a certain kind of property oftransfinitecardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least α such that α=ωα). The proposition that such cardinals exist cannot be proved in the most commonaxiomatization of set theory, namelyZFC, and such propositions can be viewed as ways of measuring how "much", beyond ZFC, one needs to assume to be able to prove certain desired results. In other words, they can be seen, inDana Scott's phrase, as quantifying the fact "that if you want more you have to assume more".[1]
There is a rough convention that results provable from ZFC alone may be stated without hypotheses, but that if the proof requires other assumptions (such as the existence of large cardinals), these should be stated. Whether this is simply a linguistic convention, or something more, is a controversial point among distinct philosophical schools (seeMotivations and epistemic status below).
Alarge cardinal axiom is an axiom stating that there exists a cardinal (or perhaps many of them) with some specified large cardinal property.
Most working set theorists believe that the large cardinal axioms that are currently being considered areconsistent with ZFC.[2] These axioms are strong enough to imply the consistency of ZFC. This has the consequence (viaGödel's second incompleteness theorem) that their consistency with ZFC cannot be proven in ZFC (assuming ZFC is consistent).
There is no generally agreed precise definition of what a large cardinal property is, though essentially everyone agrees that those in thelist of large cardinal properties are large cardinal properties.
A necessary condition for a property of cardinal numbers to be alarge cardinal property is that the existence of such a cardinal is not known to be inconsistent withZF and that such a cardinalΚ would be an uncountable initial ordinal for whichLΚ is a model of ZFC. If ZFC isconsistent, then ZFC doesnot imply that any such large cardinals exist.
A remarkable observation about large cardinal axioms is that they appear to occur in strictlinear order byconsistency strength. That is, no exception is known to the following: Given two large cardinal axiomsA1 andA2, exactly one of three things happens:
These are mutually exclusive, unless one of the theories in question is actually inconsistent.
In case 1, we say thatA1 andA2 areequiconsistent. In case 2, we say thatA1 is consistency-wise stronger thanA2 (vice versa for case 3). IfA2 is stronger thanA1, then ZFC+A1 cannot prove ZFC+A2 is consistent, even with the additional hypothesis that ZFC+A1 is itself consistent (provided of course that it really is). This follows fromGödel's second incompleteness theorem.
The observation that large cardinal axioms are linearly ordered by consistency strength is just that, an observation, not a theorem. (Without an accepted definition of large cardinal property, it is not subject to proof in the ordinary sense.) Also, it is not known in every case which of the three cases holds.Saharon Shelah has asked, "[i]s there some theorem explaining this, or is our vision just more uniform than we realize?"Woodin, however, deduces this from theΩ-conjecture, the main unsolved problem of hisΩ-logic. It is also noteworthy that many combinatorial statements are exactly equiconsistent with some large cardinal rather than, say, being intermediate between them.
The order of consistency strength is not necessarily the same as the order of the size of the smallest witness to a large cardinal axiom. For example, the existence of ahuge cardinal is much stronger, in terms of consistency strength, than the existence of asupercompact cardinal, but assuming both exist, the first huge is smaller than the first supercompact.
Large cardinals are understood in the context of thevon Neumann universe V, which is built up bytransfinitely iterating thepowerset operation, which collects together allsubsets of a given set. Typically,models in which large cardinal axiomsfail can be seen in some natural way as submodels of those in which the axioms hold. For example, if there is aninaccessible cardinal, then "cutting the universe off" at the height of the first such cardinal yields auniverse in which there is no inaccessible cardinal. Or if there is ameasurable cardinal, then iterating thedefinable powerset operation rather than the full one yieldsGödel'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 theCabal), 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 asMartin's axiom) or others that they consider intuitively unlikely (such asV = L). The hardcorerealists in this group would state, more simply, that large cardinal axioms aretrue.
This point of view is by no means universal among set theorists. Someformalists 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 thatontological 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 axiomsare restrictive, pointing out that (for example) there can be atransitive set model in L that believes there exists a measurable cardinal, even though L itself does not satisfy that proposition.
| Overview |  | |
|---|---|---|
| Axioms | ||
| Operations | ||
| ||
| Set types | ||
| Theories | ||
| ||
| Set theorists | ||
| General | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| Theorems (list) and paradoxes | |||||||||
| Logics |
| ||||||||
| Set theory |
| ||||||||
| Formal systems (list), language and syntax |
| ||||||||
| Proof theory | |||||||||
| Model theory | |||||||||
| Computability theory | |||||||||
| Related | |||||||||
