Any collection of subsets of some fixed set X, ordered by set-inclusion, is a poset; in particular, the full power set ℘(X) is a poset under set inclusion.
But whereas the full power set operation is iterated to obtain the cumulative hierarchy, the levels of the constructible hierarchy are defined strictly predicatively, that is by including at the next level only those sets which are first order definable using parameters from the previous level.
Cantor, like Dedekind, starts with the infinite set of rational numbers; and Cantor's construction again relies essentially on the full power set of the rational numbers, here in the form of arbitrary Cauchy sequences.
Here again, we start with an infinite system, that of all natural numbers; and new number systems are constructed out of it set-theoretically (although in those cases the full power set is not needed).
We can also find general models of the Peano postulates in which the universe of sets is less than the full power set of the individual universe (i.e., general models that are not absolute).
A sentence that is valid in the standard semantics is true in those general structures for which the 1-place relation universe is the full power set of the individual universe, and so forth.
(Note that by setting for some subsets, fractional partitions using arbitrary collections of sets can be thought of as fractional partitions using the full power set.) Thus, the game having a nonempty core is equivalent to its being balanced.
In his next step and proceeding further along set-theoretic and structuralist lines Dedekind introduces the set of arbitrary cuts on his initial system, thus working essentially with the bigger and more complex infinity of all subsets of the rational numbers (the full power set).
The difference is that in standard models we let plural variables range over the full power set of $D$, whereas plural variables range over a set $S$ of non-empty subsets of $D$, provided only that $S$ contains all sets definable by some formula.
It took a little under half an hour to get them all into the air, each plane needing a minute to goad its four Merlin engines up to full power before setting off down the bumpy runway.
