Large Cardinal - Hierarchy of Consistency Strength

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 A1 and A2, one of three (mutually exclusive) things happens:

1. ZFC proves "ZFC+A1 is consistent if and only if ZFC+A2 is consistent,"
2. ZFC+A1 proves that ZFC+A2 is consistent,
3. ZFC+A2 proves that ZFC+A1 is consistent.

In case 1 we say that A1 and A2 are equiconsistent. In case 2, we say that A1 is consistency-wise stronger than A2 (vice versa for case 3). If A2 is stronger than A1, then ZFC+A1 cannot prove A2 is consistent, even with the additional hypothesis that ZFC+A1 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, "s there some theorem explaining this, or is our vision just more uniform than we realize?" Woodin, however, deduces this from the Ω-conjecture, the main unsolved problem of his Ω-logic.

It should also be noted that 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.