Powerset

That is, the powerset of the powerset of S is Dedekind-finite (see below).

Her unique powerset would have greatly come in handy against an all-powerful force of evil.

An ultrafilter on ℘(ω) is Ramsey if and only if it is minimal in the Rudin–Keisler ordering of non-principal powerset ultrafilters.

For ultrafilters on a powerset ℘(S), a principal ultrafilter consists of all subsets of S that contain a given element s of S. Each ultrafilter on ℘(S) that is also a principal filter is of this form.

Given a set S, a partial ordering ⊆ can be defined on the powerset P(S) by subset inclusion, turning (P(S),⊆) into a lattice.

In summary, one can say that every complete lattice is isomorphic to the image of a closure operator on a powerset lattice.

So some of the power of comprehension was added back via the other existence axioms of ZF set theory (pairing, union, powerset, replacement, and infinity) which may be regarded as special cases of comprehension.

Such a least cut does indeed exist and one has a closure operator on the powerset lattice of all elements.

These are expensive to construct—they are usually created using the powerset construction —but are very quick to use.

This distinction is relevant in practice, but not in theory, as there exists an algorithm (the powerset construction ) that can transform any NFA into a more complex DFA with identical functionality.