Hi all, (01)
some points about ur-elements and sets. (02)
>> Isn’t the empty set a good example of something with no contact to
>> the material world?
>
> No. Empty sets abound in the real world. The set of coins in my
> pocket after I've spent the last one is the empty set. (03)
I would say "there are no coins in your pocket", rather than "there
are as many coins in your pocket as there are members in the empty set". (04)
>> Which is perhaps why someone plumped for an equally odd term
>‘Individual’.
>
> Please, don't lets go down that rat-hole again. What everyone in my
> field means by that word has NOTHING to do with sets and ur-elements. (05)
One can very easily _map_ ur-elements with individuals/particulars, if
one sees some use for this. (06)
Short passages about ur's: (07)
The term ‘ur’ comes from German, meaning ‘basic’. Ur-elements may also
be called urelemets, or simply ur’s. Ur-elements are called often
atoms, as Peter Simons explains in [Parts: A Study in Ontology, p.16]:
"An atom is an individual with no proper parts; it is accordingly indivisible
either in fact or in theory, as befits the etymology of its name. Atoms in
this strict mereological sense are not to be confused with atoms in the sense
of physics and chemistry, which may have numerous proper parts and are
far from indivisible, even in fact. Here the etymology of the name has lost
touch with progress in physics" (08)
There is a long tradition of using ur-elements, as explained by Jon
Barwise in [Admissible sets and structures, p.9]:
"In the early days of set theory, certainly in the work of Zermelo, urelements
were an integral part of the subject. The rehabilitation of urelements in
the context of admissible set theory is such a simple idea that it would be
silly to assign credit for it to any one person. Probably everyone who has
thought at all about infinitary logic and admissible sets has had a similar
idea." (09)
Collection theories that accommodate granularity can all be called set
theories. We can classify them in many ways, and here is one
classification: (010)
1.1 Empty set is accommodated
1.2 Empty set is not accommodated
2.1 Ur's are accommodated
2.2 Ur's are not accommodated (011)
There are three possible combinations (1.2 & 2.2 is not intelligible): (012)
1.1 & 2.1
1.1 & 2.2
1.2 & 2.1 (013)
If one aims to really model something physical/concrete with a set
theory, or aims to use set theory in metaphysical tool then 1.2 & 2.1
is very likely a better choice than 1.1 & 2.1 and 1.1 & 2.2. This is
because empty set more likely causes problems than solves them (1.2 &
2.1 is the same as naive set theory, at least if we are constrained on
sets with rank 1). In the cases where it is said that "empty set is
used", it appears that the use of empty set can be compensated with
something else. (014)
The use of empty set can be also constrained so that {} cannot be a
member of any set, but is used only as a marker of disjointness of two
sets. This way we do not have to worry about 'disjointness' as a
member of a set, while still maintaining the traditional use of {} as
a marker of disjointness. (015)
kind regards, (016)
-Avril (017)
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