The use of size in this blog

In this blog we talk about, mainly, three different sizes of things,
small, large and very large.
Thus
$\Set$ is the large category of small sets and functions.
$\CAT$ is the very large 2-category of large categories (and functors and natural tranformations between them).
Thus, using $`\in'$ to mean $``$is an object of$"$, we have, if $X$ is a small set,
\[X \in \Set \in \CAT.\]

$\Newextarrow{\xLongrightarrow}{3,2}{0x21D2}$

The reason for worrying about this is to try to avoid logical problems that can occur if size is ignored.
If one uses the tools of set theory without care, logical inconsistencies can occur.
For example, consider how the unrestrained use of set theoretical constructions
can lead to a contradiction
(this example returns to the normal set-theoretical meaning of $`\in'$):

1. Let $U$ (for “universe”) be the set of all sets.
2. Let $X = \{u\in U \mid u \notin u \}$.
3. Assume $X \in U$.
4. Ask the question: Is $X \in X$?
   $X \in X \xLongrightarrow{\text{defn $X$}} X \notin X$.
   $X \notin X \xLongrightarrow{\text{defn $X$}} X \in X$.
   Thus in either case we have a contradiction.

So, to avoid contradiction,
we must disallow at least one part of the reasoning in the above example.

Is the problem that we are allowing $X$ to be both a subset and an element of $U$?
No, that is generally allowable.
For example, consider the set $\bigl\{1,\{1\}\bigr\}$.
Here $\{1\}$ is both a subset (containing the first element)
and an element (the second element) of the same set.
(By the way, this situation occurs consistently in the definition of ordinal numbers,
e.g. $2 = \bigl\{\emptyset,\{\emptyset\}\bigr\}$.)

To, hopefully, avoid such problems,
in this blog we employ the hierarchy sketched above.
Then we would modify the above example to make $U$ be the set of all small sets.
We could still define $X$ as above, but it would not necessarily be in $U$,
making steps 3 and 4 invalid.

Technical note:
The term ‘small’ is relative to a chosen “Grothendieck universive”.
Also see the dichotomy small/large in Mac Lane's CWM.
We accept the “Axiom of Universes”, and use it to go a step further, to “very large” sets,
to allow for the comparison of large categories and 2-categories.