Sergei and Bill, (01)
That is the distinction between set theory and mereology: (02)
> If sets are there in the world, then no - there's nothing
> abstract going on. I'm real. Chuck is real. And the set
> of me and Chuck is real. (03)
In mereology, Bill and Chuck are each parts of the collection
that consists of Bill and Chuck. You can call that pair C,
but C is not a new entity. It is just Bill and Chuck. (04)
But in set theory, if Bill and Chuck are members of the set S,
then S is a third entity that is different from each of them.
You have three things: Bill, Chuck, and {Bill,Chuck}. You
can also construct a fourth thing, which is the set whose only
member is the set whose members are Bill and Chuck: {{Bill,Chuck}}. (05)
You can form an infinite number of distinct sets whose starting
elements are just Bill and Chuck: (06)
Bill, Chuck, {Bill}, {Chuck}, {Bill,Chuck}, {{Bill},Chuck},
{Bill,{Chuck}}, {{Bill},{Chuck}}, {{Bill}}, {{Chuck}}.... (07)
You don't even need to start with anything. You can build up
all of mathematics out of just the empty set: {}, {{}}, {{},{}},
{{{}}}, {{},{},{}}, {{{}},{{}},{{}}}... (08)
In that sense, sets are like lists in LISP. You can start with
strings, such as "Bill" and "Chuck" and create lists out of
dotted pairs ("Bill".("Chuck".())). But the dotted pair of
two strings is not a string. (09)
String concatenation is similar to mereology. The concatenation
of two strings is another string, such as "BillChuck", and there
is no way to build up anything by concatenating the empty string
to itself. No matter how many times you do so, you never get
anything other than an empty string. (010)
To answer the question about concrete vs. abstract: the infinite
number of sets you construct out of Bill and Chuck cannot be physical
because they would have an infinite amount of matter and/or energy.
Therefore, all but the bare atoms, Bill and Chuck, must be abstract. (011)
John (012)
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