>Chris,
> I thought that in Mereology the singleton mereological grouping
>(Bill) was identical to Bill. (01)
Yes. (02)
> If so, then we cannot have Bill and the mereological (Bill) as
>distinct entities. (03)
Correct. (04)
> That is
>How mereology differs from sets, no? (05)
Indeed. (06)
> Have I misunderstood? (07)
No. (08)
Pat(-2) (09)
>
>Pat
>
>
>-----Original Message-----
>From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx
>[mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of
>Christopher Menzel
>Sent: Thursday, February 08, 2007 12:11 AM
>To: [ontolog-forum]
>Subject: Re: [ontolog-forum] Visual Complexity
>
>> That is the distinction between set theory and mereology:
>>
>>> 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.
>>
>> 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.
>
>John, that is not correct. The mereological sum of Bill and Chuck --
>call it Bill+Chuck -- is typically defined in mereology as the
>smallest thing that has Bill and Chuck as parts. (Equivalently, it
>is the unique thing X such that anything that overlaps X either
>overlaps Bill or overlaps Chuck.) It is not "just Bill and Chuck".
>It is a third thing distinct from the two of them.
>
>> 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}.
>
>And in mereology you have 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}}.
>
>Now THERE is where mereology and set theory part company. Mereology
>does not support an unbridled hierarchy of sums parallel to the
>hierarchy of sets in set theory. As you note:
>
>> You can form an infinite number of distinct sets whose starting
>> elements are just Bill and Chuck:
>>
>> Bill, Chuck, {Bill}, {Chuck}, {Bill,Chuck}, {{Bill},Chuck},
>> {Bill,{Chuck}}, {{Bill},{Chuck}}, {{Bill}}, {{Chuck}}....
>>
>> You don't even need to start with anything. You can build up
>> all of mathematics out of just the empty set: {}, {{}}, {{},{}},
>> {{{}}}, {{},{},{}}, {{{}},{{}},{{}}}...
>
>By contrast, in mereology, the sum of Bill and Bill+Chuck is just Bill
>+Chuck; likewise, the sum of Bill's left arm and Bill is just Bill.
>In set theory, as you note, you get the distinct entities {Bill,
>{Bill, Chuck}} and {BillsLeftArm, Bill}. But in mereology and set
>theory alike, the sum/set of Bill and Chuck is a third thing distinct
>from Bill and Chuck.
>
>-chris
>
>
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