A question about the relative sizes of Measurable and Supercompact Cardinal Numbers.

Question

Is the least Supercompact Cardinal Number necessarily greater than the least Measurable Cardinal Number?

Answer 1

Supercompactness is much stronger than measurability:

Supercompactness of $\kappa$ states that for any $\mu$, we can find embeddings $j:V\to M$ of the universe of sets with $M$ transitive, critical point $\kappa$ and $M$ closed under $\mu$ sequences. This implies in particular that for any set $x\in V$, we can find one such embedding $j$ with $x\in M$ (that is to say, supercompact cardinals are strong cardinals. But in fact supercompactness is a significantly... ehm... stronger requirement than strongness).

In particular, given any non-principal $\kappa$ complete ultrafilter $\mathcal U$ on $\kappa$, we can assume that $\mathcal U\in M$. Standard reflection arguments (which are a direct consequence of the elementarity of $j$) give us that the set of measurable cardinals below $\kappa$ is stationary in $\kappa$, and much more. (So if $\kappa$ is strong, then it is the $\kappa$-th measurable cardinal, and is the $\kappa$-th cardinal $\lambda$ such that $\lambda$ is the $\lambda$-th measurable cardinal, etc.)

That said, there is a small subtlety: It is (widely?) expected that supercompactness is equiconsistent with strong compactness, and it is known since work on Magidor in the 70s that it is consistent that the first strongly compact cardinal is the first measurable. (Nevertheless, if there strongly compact cardinals at all, we can find inner models with many measurable cardinals, so Magidor's "identity crisis" theorem is not a result about the consistency strength of these assumptions.)