Editing Large cardinal (section)

==Hierarchy of consistency strength==
A remarkable observation about large cardinal axioms is that they appear to occur in strict [[linear order]] by [[consistency strength]]. That is, no exception is known to the following: Given two large cardinal axioms ''A''<sub>1</sub> and ''A''<sub>2</sub>, exactly one of three things happens:
#Unless ZFC is inconsistent, ZFC+''A''<sub>1</sub> is consistent if and only if ZFC+''A''<sub>2</sub> is consistent;
#ZFC+''A''<sub>1</sub> proves that ZFC+''A''<sub>2</sub> is consistent; or
#ZFC+''A''<sub>2</sub> proves that ZFC+''A''<sub>1</sub> is consistent.
These are mutually exclusive, unless one of the theories in question is actually inconsistent.

In case 1, we say that ''A''<sub>1</sub> and ''A''<sub>2</sub> are [[Equiconsistency|equiconsistent]]. In case 2, we say that ''A''<sub>1</sub> is consistency-wise stronger than ''A''<sub>2</sub> (vice versa for case 3). If ''A''<sub>2</sub> is stronger than ''A''<sub>1</sub>, then ZFC+''A''<sub>1</sub> cannot prove ZFC+''A''<sub>2</sub> is consistent, even with the additional hypothesis that ZFC+''A''<sub>1</sub> is itself consistent (provided of course that it really is). This follows from [[Gödel'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?" [[W. Hugh Woodin|Woodin]], however, deduces this from the [[Ω-conjecture]], the main unsolved problem of his [[Ω-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 a [[huge cardinal]] is much stronger, in terms of consistency strength, than the existence of a [[supercompact cardinal]], but assuming both exist, the first huge is smaller than the first supercompact.