Matthew, John, Chris, (01)
John already told it, but to clarify once more, when you have only
rank 1 sets, and you decide that you do not use the memberOf operator,
and you forget the atoms that are not members of sets, then you can
apply sets and discrete/atomistic mereology in an identical way. But
once I suggest this, people immediately say "what about a in {a}?"
This shows that unnecessary elements should not be included, only
because they are a potential source of confusion. In general, the less
unnecessary confusion, the better. That's why it's probably better to
follow John's advise and use 'set' only to refer to Cantor-originated
stuff. But all Cantor-originated stuff does not have to suffer from
the problems that have been pointed out by a long parade of the
greatest philosophers in this planet. (02)
I'm only interested in structural features of set theory, i.e., about
applying set theoretic structures to the concrete measurable reality.
I'm not talking about set theory as a foundation of mathematics at
all. Having this view, only a very diminished version of set theory is
required. Here is one version, which I call finitist set theory (FST): (03)
http://www.cs.helsinki.fi/u/astyrman/fst.pdf (04)
It is currently under evaluation. From the viewpoint of a
mathematician who is concentrated on ZF(C), this is just an
unimportant and uninteresting theory that can be encoded by using ZF.
From a structural point of view, it is the minimal theory that can be
used in building granular structures, and it has several advantages
compared to ZFU and KPU (set theories which accommodate ur-elements). (05)
1. It is finite, i.e., there are no transfinite sets. You explicitly
assign the number of atoms (ur-elements) and the maximum rank. Thus,
axiom of foundation is not needed: it is needed only to exclude some
implications of the transfinite hierarchy, but because there are only
finitely many sets, there is no danger of non-wellfounded structures
in the first place. (06)
2. The disclusion of empty set is inherited from mereology. That is a
great simplification: empty set is not needed, and thus there is no
use to have it. It only messes up conceptual modeling. (07)
3. All set theories I'm aware of (except FST) have copied the axiom of
union from ZF. It follow from ZF's union that in order to have all
rank n sets, also rank n+1 sets have to exist. In FST, you define the
maximum rank, and the axioms give all sets from rank 1 to n. This was
done by modifying the axiom of union. Also the axiom of pairing was
diminished into the axiom of singleton sets. Pairing is just
unnecessarily strong. (08)
FST should be seen as a nonproblematic foundation of granularity. In
general, you can encode whatever with ZFC. Then again, if you only
need the encoded version, the question rises that for what purpose do
you need ZFC. In the big picture, you have natural language and you
can make any theory with it. So, when talking about modeling the
measurable reality, why do we need to have ZFC as a mid-layer between
NL and e.g. FST? Scientists should not be concentrated too much on
ZFC. If someone needs granularity, she can have it, without having to
learn ZFC. Take a biologist, chemist, a computer scientist, or anyone
who is interested in modeling the measurable reality. It's not
necessary for them to spend time on learning and undestanding ZFC:
they can enjoy the fruitful features of set theory without ZFC. (09)
-Avril (010)
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