Let $(I,<)$ be a directed, partially ordered set. Consider an inverse system $(S\_i)\_{i \in I}$ of finite sets, i. e. a functor $S:I^{op}\to \mathbf{FinSet}$. What is the maximum possible cardinality of the inverse limit $A=\varprojlim S$ of this system? It is obvious that $|A|=2^{\aleph\_0}$ can be achieved (for example by taking $I=\Bbb N$ and $S\_i=\mathbb Z/p^i\mathbb Z$ yielding $A=\Bbb Z\_p$), but I don't see how to do any better.



Maximum cardinality of a filtered limit of finite sets
You can get arbitrarily large cardinalities. For instance, let $X$ be any set and consider the poset $I$ of finite partitions of $X$, ordered by refinement. There is a "tautological" filtered system of finite sets indexed by $I$, and $X$ naturally maps to it. Clearly the map from $X$ to the limit of this system is injective (in fact, the limit can naturally be identified with the set of ultrafilters on $X$, with the natural map being the inclusion of principal ultrafilters).


More generally, you can do a similar construction with the poset of (finite) partitions of any Boolean algebra, and the limit will be the Stone space. Conversely, any inverse limit of finite sets acquires a natural profinite topology. If all the maps in the original inverse system were proper (i.e., non-bijective) surjections, then it can be considered as a cofinal subsystem of the system of all clopen partitions of the limit. That is, every inverse system of proper surjections of finite sets is cofinal in a system of this form. Furthermore, any inverse system of finite sets can be canonically transformed into one in which the maps are proper surjections without changing the limit: first remove all elements of $S\_i$ that are not in the image of every map $S\_j\to S\_i$, and then impose an equivalence relation on $I$ that identifies $i$ and $j$ whenever $S\_j\to S\_i$ is a bijection.


For example, in Joel's construction, the profinite topology is the topology that sees $X$ as the one-point compactification of the discrete topology on $X\setminus\{0\}$. The Boolean algebra of clopen subsets can be identified with the finite-cofinite Boolean algebra on $X\setminus\{0\}$, and the inverse system is isomorphic to a (cofinal) system of partitions of this algebra: given a finite subset $\{0,x\_1,\dots,x\_n\}\subseteq X$, identify it with the partition $\{x\_1\},\{x\_2\},\dots,\{x\_n\},X\setminus \{0,x\_1,\dots,x\_n\}$.


