Zero Sharp - Consequences of Existence and Non-existence

Consequences of Existence and Non-existence

Its existence implies that every uncountable cardinal in the set-theoretic universe V is an indiscernible in L and satisfies all large cardinal axioms that are realized in L (such as being totally ineffable). It follows that the existence of 0# contradicts the axiom of constructibility: V = L.

If 0# exists, then it is an example of a non-constuctible Δ1
3 set of integers. This is in some sense the simplest possibility for a non-constructible set, since all Σ1
2 and Π1
2 sets of integers are constructible.

On the other hand, if 0# does not exist, then the constructible universe L is the core model—that is, the canonical inner model that approximates the large cardinal structure of the universe considered. In that case, Jensen's covering lemma holds:

For every uncountable set x of ordinals there is a constructible y such that x ⊂ y and y has the same cardinality as x.

This deep result is due to Ronald Jensen. Using forcing it is easy to see that the condition that x is uncountable cannot be removed. For example, consider Namba forcing, that preserves and collapses to an ordinal of cofinality . Let be an -sequence cofinal on and generic over L. Then no set in L of L-size smaller than (which is uncountable in V, since is preserved) can cover, since is a regular cardinal.