Chris and Avril, (01)
CM
> virtually every text on set theory presents the axioms with just a
> single binary predicate "∈" for membership. The subset relation is
> always defined; using "isin" instead of "∈": (02)
I agree. But the theory of Boolean collections is much simpler
and has much less structure than Cantor's set theory. If you
consider the lattice of all possible first-order theories, then
both set theory and many versions of mereology can be considered
specializations of the theory of Boolean collections. (03)
With Cantor's elementOf operator, you have an enormous amount of
power for building up arithmetic, the hierarchies of infinity, etc. (04)
Lesniewski considered that to be too much power. He claimed that
it is clearer and simpler to start with mereology, which cannot,
by itself, be used to construct arithmetic. (05)
If you want to define integers, you can add the successor function
plus Peano's axioms. The integers are a very widely used basis
for constructing other kinds of countable models. (06)
AS
> If we consider only sets with rank 1, that is, no inner sets, then
> mereology and set theory become identical. (07)
CM
> That's a pretty misleading way to put it as the membership relation
> in set theory does not correspond to anything in mereology, so there
> will be many truths of set theory that don't correspond to anything
> in mereology, e.g., "a ∈ {a}", "a ≠ {a}", etc. (08)
I agree with Chris. But I would add that some programming
languages that support things they call sets define "a = {a}". (09)
A better way to state Avril's point is that the theory of Boolean
collections is sufficient for practical applications in computer
science. (010)
Note that most major programming languages support lists (including
lists of lists). Some languages support sets by ignoring the implicit
ordering of lists. But very few programmers use that option because
it tends to be very inefficient. (011)
Relational databases define relations as sets of tuples (lists),
but they never use sets of sets. Therefore, Boolean collections
would be sufficient to support SQL and other practical applications. (012)
Has anyone ever found an application or even a contrived example
in computer science that would require uncountable sets? (013)
John (014)
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