It’s quite standard nowadays to characterise the measurable cardinals as the cardinals such that there exists a normal -complete non-principal measure on . As we continue climbing the large cardinal hierarchy we get to the strong cardinals, Woodin cardinals and superstrong cardinals, all of which are characterised by extenders, which can be viewed as particular sequences of normal measures on . This trend then stops, and there’s a shift from measures on to measures on , being the set of subsets of of cardinality less than . Now, how does one work with such measures? Where are the differences between our usual measures and these kinds? And how can we view this shift as expanding the amount of things that we can measure?
Before we start, let’s note that the step from measures on to measures on is a simple matter of generalisation. Because if is a non-principal measure on ,then we can associate to a filter on by setting
Note that non-principality implies that is a measure as well and it furthermore satisfies that whenever , the set of all satisfying that is in ; we say that is fine. Now, if we started off with a fine measure on then we could’ve defined as
Note here fineness of ensures that is a non-principal measure on , showing the equivalence between the existence of these two different types of measures. We furthermore see that -completeness for is equivalent to -completeness for . This leads us to the definition of strong compactness.
Definition. Let be cardinals. Then is -strongly compact if there exists a -complete fine measure on . We also simply say that is strongly compact if it’s -strongly compact for all .
The above argument then shows that being measurable is equivalent to it being -strongly compact, and considering larger then gives us a natural way to improve the strength of measurability. This is somehow orthogonal to the extender approach, as that approach ensures that we can measure subsets of in many different (coherent) ways, and this other approach is instead about measuring subsets of cardinals greater than .
Recall that a measure on is normal if for every -sequence of measure one sets, the diagonal intersection has measure one as well, where
Analogously, normality of a measure on is precisely the same, except that we define the diagonal intersection slightly differently. If we let be a -sequence of measure one sets, then we set
and again is normal if for every .
Now note that, in the case, the normality of is equivalent to normality of , in the sense of the above. The argument uses -completeness, which we may assume since normality of a measure on does imply -completeness (here it’s important that we’re looking at -sized subsets). We arrive at supercompacts.
Definition. Let be cardinals. Then is -supercompact if there exists a normal fine measure on . We also simply say that is supercompact if it’s -supercompact for all .
As the existence of a normal measure on is equivalent to the existence of a -complete one, the above argument then also shows that measurability is also equivalent to being -supercompact! As we increase however, -supercompactness diverges from -strongly compactness.
Taking a step back, we can also start talking about club and stationary subsets of , in complete analogy with the usual terminology. We say that a subset is closed if it’s closed under unions, and unbounded if for every there’s a such that . is then club if it’s both closed and unbounded, and a subset is stationary if it meets all clubs. If we then look at the collection of stationary sets of this suddenly makes an appearance in the realm of Woodin cardinals, as the stationary tower forcing.