Put more simply but less carefully, for any set of parts of hypergunk there is another set of parts of hypergunk such that the second set has a strictly greater cardinality than the first.
The possibility of hypergunk makes all sorts of trouble for views like Lewis's, and others as well. (Want to hold that worlds are maximal sets of propositions? There are no such sets if hypergunk is possible.) Or so argues Daniel Nolan in his "Classes, Worlds and Hypergunk".
If hypergunk is possible, then there are possibly more than set-many individuals. This is because there is no set that has as members all the parts of hypergunk. For suppose there were such a set and its cardinality were c (for reductio). Then by the definition of hypergunk, there is another set with a cardinality c* such that c* > c. But since it's not the case that c > c, our assumption that there is a set that has as members all the parts of hypergunk is false. So there is no such set.
Some immediate implications for Lewis's view:
- There are worlds that do not correspond to any set of individuals
- There is no set of all worlds
- There are no necessary truths
- There are no truths about some worlds
- There are no truths about some individuals
(Former Priestly seminarians may also be interested in these papers by Nolan.)