The relationship between the Singular Cardinal Hypothesis, Jensen's square
principle, very good scales and large cardinals is important in singular
cardinal arithmetic and in understanding how much the universe resembles L.
Jensen showed that square holds in L. On the other hand, weak square fails
above a supercompact, and implies that every scale is good.
There have also been results about singular cardinals that are not relative
consistency results. Using PCF theory Shelah showed that if 2^aleph_n < aleph_omega for every n < omega, then 2^aleph_omega < aleph_(omega_4).
Scales are a central concept in PCF theory and are very
useful in exploring the tension between combinatorial principles like square
and the reflection properties in the presence of large cardinals.
We will discuss relative consistency results about the relationship between
these principles in the context of forcing and large cardinals.